A GEOMETRIC APPROACH TO ELLIPTIC CURVES WITH TORSION GROUPS Z / 10 Z , Z / 12 Z , Z / 14 Z , AND Z / 16 Z

. We give new parametrisations of elliptic curves in Weier-strass normal form y 2 = x 3 + ax 2 + bx with torsion groups Z / 10 Z and Z / 12 Z over Q , and with Z / 14 Z and Z / 16 Z over quadratic ﬁelds. Even though the parametrisations are equivalent to those given by Kubert and Rabarison, respectively, with the new parametrisations we found three in-ﬁnite families of elliptic curves with torsion group Z / 12 Z and positive rank. Furthermore, we found elliptic curves with torsion group Z / 14 Z and rank 3–which is a new record for such curves –as well as some new elliptic curves with torsion group Z / 16 Z and rank 3.


Introduction
An elliptic curve E over a field K is a smooth projective curve of genus 1 equipped with a K-rational point. When embedded in the affine plane, E is described by the Weierstrass model y 2 + a 1 xy + a 3 y = x 3 + a 2 x 2 + a 4 x + a 6 , where the coefficients belong to K. Elliptic curves can be represented by several other equations. The interested reader may consult [14,Ch. 2]. In the few past decades, many alternative equations describing E have been introduced in the context of cryptographic applications. In homogeneous coordinates, the curve y 2 = x 3 + ax 2 + bx becomes Γ : Y 2 Z = X 3 + aX 2 Z + bXZ 2 .
With respect to the curve Γ α,β,γ,δ , we can compute the conjugate of a point by the following: Fact 3. Let P = (x 0 , y 0 ) be a point on Γ α,β,γ,δ . Then Proof. Let P = (x 0 , y 0 ) be a point on Γ α,β,γ,δ . Then which implies that x 0 is a root of and since the other root is α+δ( Let Γ a,b : y 2 = x 3 + ax 2 + bx be a regular curve over some field F with torsion group Z/2nZ (for some n ≥ 5). Each element of the group Z/2nZ = {0, 1, . . . , 2n − 1} corresponds to a point on Γ a,b . LetT be the unique point of order 2. ThenT corresponds to n. Furthermore, let A ′ be a point on Γ a,b which corresponds to 1. Then A ′ is of order 2n. Finally, let B ′ be the point on Γ a,b which corresponds to 2. Then A ′ + A ′ = B ′ . Now, by Lemma 2, there is a projective transformation Φ which maps the curve Γ a,b to the curve Γ α,β,γ,δ , the point A ′ to the point A = (1, 1), and the pointĀ ′ to the pointĀ = (−1, −1). Moreover, since A +Ā corresponds to 1 + (n + 1) = n + 2, we obtain that A +Ā =B. In other words, A #Ā = −B, which implies that −B is on the line AĀ. Hence, −B = (u, u) for some u ∈ F, and therefore B = (v, u) for some v ∈ F.
Since the points A,Ā, B,B belong to the curve Γ α,β,γ,δ , we obtain By applying Φ −1 to the curve Γ u,v , we obtain the curve Γ a,b with In the following sections we shall apply this approach to elliptic curves with torsion groups Z/2nZ for n = 5, 6, 7, 8.

Elliptic Curves with Torsion Group Z/10Z
To warm up, we give a parametrisation of elliptic curves with torsion group Z/10Z.
Let Γ a,b : y 2 = x 3 + ax 2 + bx be a regular curve with torsion group Z/10Z over Q. Each element of the group Z/10Z = {0, 1, . . . , 9} corresponds to a rational point on Γ a,b . LetT be the unique point of order 2. ThenT correspond to 5. Furthermore, letÃ andB be the rational points on Γ a,b which correspond to 1 and 2, respectively. ThenÃ is of order 10 andB is of order 5. Finally, let Φ be a projective transformation Φ which maps the curve Γ a,b to the curve Γ α,β,γ,δ , the pointÃ to the point A = (1, 1), and the conjugate ofÃ to the pointĀ = (−1, −1). Let B := Φ(B) and T := Φ(T ). Then, for A, −A,Ā, . . . we obtain the following correspondence between these points on Γ α,β,γ,δ and the elements of the group Z/10Z: By definition, we have: (i) The points A,Ā, −B are collinear.
(ii) The points A, B,B are collinear.
Then, the curve Γ a 1 ,b 1 : is an elliptic curve with torsion group Z/10Z. Conversely, if Γ a,b is a regular elliptic curve with torsion group Z/10Z, then there exists a u ∈ Q such that Γ a,b is isomorphic to Γ a 1 ,b 1 .

Remarks.
• In [6, Table 3, p. 217], Kubert gives the following parametrisation of curves of the form with torsion group Z/10Z (see also Kulesz [8,p. 341,(1.1.9)], who found Kubert's parametrization in a different way): After transforming Kubert's curve into the form Now, by substituting inã andb the values p and q with p + q and 2q, respectively, and setting u = p q , we obtain 4a 1 and 16b 1 , respectively, which shows that the two parametrisations are equivalent.
• Recall that the Calkin-Wilf sequence s 1 = 1, s n+1 = 1 2⌊s n ⌋ − s n + 1 lists every positive rational number exactly once. By checking the first 22 000 fractions in this sequence we found, with the help of MAGMA, 46 elliptic curves with torsion group Z/10Z and rank 3.
• The following table gives the fractions p/q and their indices in the Calkin-Wilf sequence of six of the 25 known elliptic curves with torsion group Z/10Z and rank 4 (see [2]): MAGMA computations reveal three generators on the curve and confirm that its root number is 1, therefore the rank should be even. The point search up to height 2 38 on each of the 256 4-coverings for each of the 4 curves in the isogeny class did not uncover the missing last generator. We leave it as an open challenge to test new descent methods.

Elliptic Curves with Torsion Group Z/12Z
Let us now consider parametrisations of elliptic curves with torsion group Z/12Z. By similar arguments as above, one can show the following result.
Theorem 5. Let t ∈ Q \ {1} be a positive rational and let Then the curve Γ a 1 ,b 1 : is an elliptic curve with torsion group Z/12Z. Conversely, if Γ a,b is a regular elliptic curve with torsion group Z/12Z, then there exists a positive rational t such that Γ a,b is isomorphic to Γ a 1 ,b 1 . Table 3, p. 217], Kubert gives the following parametrisation of elliptic curves of the form with torsion group Z/12Z (see also Kulesz [8,p. 341,(1.1.10)], who found Kubert's parametrisation in a different way): After transforming Kubert's curve into the form Now, for t = r s we obtain a 1 := 6r 8 + 48r 6 s 2 + 12r 4 s 4 − 2s 8 and Then, by substituting inã andb, r with r + s and s with 2s, we obtain 4a and 16b, respectively. This shows that the two elliptic curves and Γ a,b : y 2 = x 3 + ax 2 + bx are equivalent.

Elliptic Curves of Rank at Least 2
By checking the first 3441 fractions r/s of the Calkin-Wilf sequence we found, with the help of MAGMA, 125 fractions which lead to elliptic curves with torsion group Z/12Z and rank 2, and among these 3441 fractions, we even found some which lead to curves with rank 3.
As a matter of fact, we would like to mention that until today (November 2021), up to isomorphisms only one elliptic curve of rank 4 is known, namely discovered by Fisher in 2008 (see [2]). This curve is isomorphic to

Families of Elliptic Curves with Positive Rank
In this section, we construct three infinite families of elliptic curves Γ a,b with positive rank. Other such families were found, for example, by Rabarison [11,Thm. 12], Kulesz [8,Thm. 2.12] (see also [7,Sec. 2.12]), and by Suyama (see [9, p. 262 f]). Although the parametric families of positive rank and torsion group Z/12Z are not explicitly given in the work of Rabarison, they are mentioned on page 17, line 3 of his manuscript and on page 90, line 1 of his thesis. In fact, the elliptic curves which correspond to our three families are, in Cremona's notation, the curves 368d1, 226a1 and 720e2.
The finite torsion points of Γ t are given in the following table: Order x-coordinate y-coordinate Now, if we find any additional rational point P on Γ t , then the order of P is infinite which implies that the rank of Γ t is positive. On the other hand, if we find an infinite family T of values for t such that for every t ∈ T , the curve Γ t has an additional point, then {Γ t : t ∈ T } is an infinite family of elliptic curves with torsion group Z/12Z and positive rank.

First Family
Then P 1 is a rational point on Γ t if and only if v 2 = −(t 4 + 8t 3 + 2t 2 + 1) for some rational v.
This quartic curve has a rational solution (t, v) = (−1, 2), hence, by [4, p. 472], it is equivalent to the elliptic curve which is a rank-1 elliptic curve, where G 1 := (1, 1) is a point of infinite order. In particular, for all but finitely many k ∈ Z, for [k]G 1 = (x k , y k ) and t k := (y k − 1)/(2x k − y k − 1), Γ t k is a non-singular curve with torsion group Z/12Z and positive rank. where Γ a,b has rank 2. In order to compute the rank of Γ a,b , it seems to be faster to use Kubert's form (1) with τ = (r + s)/(2s) where t 2 = r/s, which gives us The following table summarizes what we have found with the help of MAGMA. MAGMA calculations also confirm the rank to be at least 1 for both k = 5 and k = 6.
Then P 2 is a rational point on Γ t if and only if v 2 = t 4 − 2t 3 + 13t 2 + 4t + 4 for some rational v.
This quartic curve has a solution (t, v) = (0, 2) and, by [14,Thm. 2.17], is birationally equivalent to which is, according to MAGMA, a rank-1 elliptic curve, where G 2 := (0, 1) is a point of infinite order. In particular, for all but finitely many k ∈ Z, for [k]G 2 = (x k , y k ) and t k := 2(x k + 2)/(y k + 1), Γ t k is a non-singular curve with torsion group Z/12Z and positive rank. MAGMA calculations also confirm that the rank is exactly 1 for k = 7 and k = 10, and the rank is at least 1 for k = 8. This quartic curve has a solution (t, v) = (1, 12), hence it is equivalent to an elliptic curve

Third Family
of rank 1, generated by the point G 3 = (−5, −18) of infinite order, as determined by MAGMA. For all but finitely many k ∈ Z, for [k]G 3 = (x k , y k ) and t k := (y k − 3x k + 3)/(y k + 9x k + 63), Γ t k is a non-singular curve with torsion group Z/12Z and positive rank.
We have computed the rank of the curve Γ t k for k = −2, 2, 3, 4:

Elliptic Curves with Torsion Group Z/14Z
For an elliptic curve over some field F with torsion group Z/14Z, starting with a value for u ∈ F, we compute a value for v, which will lead to a parametrisation of elliptic curves with torsion group Z/14Z. In homogeneous coordinates, we obtain and therefore Since the point C belongs to g, we must have the scalar product g, C = 0, i.e., belong to F, and hence 1 − 2u + u 2 + 4u 3 belongs to F.
On the other hand, if Γ a,b is an elliptic curve over F with torsion group Z/14Z, then we can transform this curve (over F) to the curve Γ α,β,γ,δ which contains the points A = (1, 1), B = (v, u), and −B = (u, u) with the above properties. In particular, we have that u, v ∈ F which implies that 1 − 2u + u 2 + 4u 3 belongs to F. q.e.d.
As an immediate consequence we obtain:

High Rank Elliptic Curves with Torsion Group Z/14Z
By some further calculations, we can slightly simplify the formulae for the parameters a and b of the curve Γ a,b . For z = 1 − 2u + u 2 + 4u 3 we have: , the corresponding curve has rank 2. We would like to mention that different values of u ∈ Q may not necessarily lead to different quadratic fields Q( d). For example, for u 1 = 4/9 and u 2 = 5/13, both curves have torsion group Z/14Z over the same quadratic field Q( √ 265). Moreover, for u 1 and u 2 , the corresponding curves have the same rank, which follows from the following result (see Rabarison [12,Lem. 4.4]).

Fact 8. Let d be a square-free integer and let
Then the elliptic curve has torsion group Z/6Z over F d for d = −7 and has torsion group Z/2Z × Z/6Z over F −7 .
Now, since (u 0 , z) is a non-torsion point on Γ 0 , the curve Γ 0 has rank ≥ 1 over F d , and by adding (in the case d = −7) the 6 torsion points of Γ 0 to (u 0 , z), we obtain the following 6 values for u, which all lead to essentially the same curve with torsion group Z/14Z over F d : For example, if u 1 = 4/9, then u 2 = 5/13 and the corresponding curves are essentially the same.
In order to obtain different curves with torsion group Z/14Z, we can, for example, start with an arbitrary u 0 ∈ Q \ {−1, 0, 1} and double the point (u 0 , z) on Γ 0 (over the corresponding field F d ). This way, we get the following value for u: For example, taking u 0 = 1/2 (curve of rank 0) we produce a different Z/14Z curve with u = −11/12 (rank 1) over the same field Q( 3). Similarly, taking u 0 = 2/3 (curve of rank 1) we produce a different Z/14Z curve with u = −8/15 (rank 0) over the same field Q( 105).
Another approach to find independent values for u would be to search for values d, such that Γ 0 has high rank over F d . With the help of MAGMA we found an abundance of quadratic fields F d over which the Z/6Z curve Γ 0 has rank 2-5. The fields F d with the smallest absolute d-values for the curve Γ 0 of rank 2, 3, 4, 5 are F 22 , F 874 , F −5069 , F 1578610 , respectively. Two essentially different Z/14Z curves over Q( 22) we found this way are produced by u 1 = 1/8 and u 2 = 7/4. The former curve has rank 0, whereas the latter has rank 1. The mentioned u-values correspond to the two generators of Γ 0 , which has rank 2 over F 22 . Similarly, the four Z/14Z curves over Q( 2233) produced by u 1 = 4/7 (curve of rank 2), u 2 = 13 (rank 1), u 3 = 1/28 (rank 0), and u 4 = −2/11 (rank 1) are all essentially different, as all the u-values correspond to pairwise linearly independent combinations of the three generators of the curve Γ 0 over F 2233 .
Let us now turn back to the search of high rank elliptic curves with torsion group Z/14Z. With the help of MAGMA we first found that for u = 11/5 (or equivalently for u = −3/8), the produced curve has rank 3 -the current record for torsion group Z/14Z (see Dujella [3]). The curve in Weierstrass normal form is and is isomorphic to the curve with the three independent points of infinite order Later, we found that also u = Another parametrisation of elliptic curves over a quadratic field with torsion group Z/14Z is given by Rabarison [12,Sec. 4.2]. The defining polynomial of the quadratic field is w 2 +wu+w = u 3 − u, which leads to The parametrised curve is Eã ,b : y 2 +ãxy +by = x 3 +bx 2 where the point (0, 0) is a point of order 14.
Like Kubert's parametrisation for elliptic curves with torsion group Z/10Z or Z/12Z, Rabarison's parametrisation for elliptic curves with torsion group Z/14Z can be transformed to our parametrisation. For this, notice first that a and b depend on u and v. Now, for we obtain expressions for a and b which just depend on u and z. On the other hand, if we set then, for we obtain exactly the same expressions for a and b (also depending just on u and z). A similar result we get forã andb by setting With respect to Rabarison's parametrisation, the curve given above with rank 3 obtained by u = 11/5 is

A Normal Form for Elliptic Curves with Torsion Group Z/14Z
For u ∈ Q, d : over the quadratic field Q( √ d) has torsion group Z/14Z. Notice that Γ u,v has two points at infinity, namely (0, 1, 0), which is the point of order 1, and (1, 0, 0), which is the point of order 2. The finite torsion points of Γ u,v are given in the following table.
Order x-coordinate y-coordinate As a matter of fact we would like to mention that for where the point at infinity (1, 0, 0) is moved to (0, 0).

Elliptic Curves with Torsion Group Z/16Z
In this section we use our geometric approach to construct a parametrisation of elliptic curves with torsion group Z/16Z.
Proof. Assume that the curve Γ α,β,γ,δ over F has torsion group Z/16Z and that for the points A andĀ, which correspond to 2 and 10, respectively, we have A = (1, 1) andĀ = (−1, −1). Furthermore, let B and D be points on Γ α,β,γ,δ which correspond to 7 and 4, respectively. Then, the group table with respect to these points, their inverses and their conjugates, is given by the following Because A andĀ are on Γ α,β,γ,δ , we have β = 1 and δ = −(α + γ). Now, since A #Ā = D, the point D is on the line passing through A andĀ, and therefore, D = (x 0 , x 0 ) for some x 0 ∈ F. Since D # D = T , this implies that the tangent to the curve Γ α,β,γ,δ with contact point D is parallel to the x-axis, which gives us Furthermore, we have −D =D, which implies that forD = (x, −x 0 ) we have x 0 =x 0 , where by Fact 3,x Thus, by (2) we obtain and for a point P = (x, y) on Γ α,β,γ,δ , the x-coordinate ofP is Now, let us consider the points B andB. Since B #B =Ā, the pointĀ is on the line g passing through B andB. Let λ be the slope of g, then g(x) = λx + (λ − 1) .
Because the line g passes through the the points B = (x 1 , y 1 ) andB = (x 1 , −y 1 ), we must have g(x 1 ) = −g(x 1 ), and solving this equation for λ gives . Furthermore, we must have that the points B andB are on Γ α,β,γ,δ , i.e., which finally gives us the following four values for x 1 : Since x 1 and α belong to F, this implies that at least one of belongs to F. On the other hand, if at least one of z 1 , z 2 , z 3 , z 4 belongs to F, then also the corresponding x 1 belongs to F, which shows that there exists an elliptic curve Γ a,b with torsion group Z/16Z over F. q.e.d.
As a consequence of Theorem 9 we obtain Corollary 10. Let m ∈ Q \ {−1, 0, 1} and let Then for each i ∈ {1, 2} there is an elliptic curve over Q( d i ) with torsion group Z/16Z.

High Rank Elliptic Curves with Torsion Group Z/16Z
In [12, p. 38], Rabarison listed a single Z/16Z curve of rank 1 over Q( √ 10), and in [1], an example of a curve of rank 2 over Q( √ 1 785) is provided. In [10], Najman used the 2-isogeny method to construct a Z/16Z curve over Q( √ 34 720 105) of proven rank 3 and conditional rank 4, starting from a rank-3 curve with the torsion group Z/2Z × Z/8Z. The three curves mentioned above can be reproduced by using formula (3) for m = 3, m = 4, and m = 12/17, respectively.
As [3] lists only the smallest, by magnitude, d-values for quadratic fields Q( √ d), the third author has shown that a Z/16Z curve of conditional rank 4 can be built over Q( √ 17 381 446) by using m = 29/65 in formula (3). This is a current record for Z/16Z curves of conditional rank 4. Uncovering generators on the d-twists to prove rank 4 unconditionally for Z/16Z curves over Q( √ d) remains a challenge. The only MAGMA calculation that has not resulted in a crash corresponds to the curve with m = 12/17. After successfully performing both 8-descent and 3-descent in 68 hours, no generator was found on the quadratic twist.