Genus 2 curve 13689.a.13689.1

• Label: 13689.a.13689.1
• Conductor: $13689$
• Discriminant: $13689$
• Mordell-Weil group: \(\Z \oplus \Z\)
• Sato-Tate group: $E_6$
• \(\End(J_{\overline{\Q}}) \otimes \R\): \(\mathrm{M}_2(\R)\)
• \(\End(J_{\overline{\Q}}) \otimes \Q\): \(\mathrm{M}_2(\Q)\)
• \(\End(J) \otimes \Q\): \(\mathsf{CM}\)
• \(\overline{\Q}\)-simple: no
• \(\mathrm{GL}_2\)-type: yes

Minimal equation

$y^2 + (x^3 + x + 1)y = x^5 - x^4 - 4x^3 - 2x^2$

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(13689\)	\(=\)	\( 3^{4} \cdot 13^{2} \)
Discriminant:	\( \Delta \)	\(=\)	\(13689\)	\(=\)	\( 3^{4} \cdot 13^{2} \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(516\)	\(=\)	\( 2^{2} \cdot 3 \cdot 43 \)
\( I_4 \)	\(=\)	\(8073\)	\(=\)	\( 3^{3} \cdot 13 \cdot 23 \)
\( I_6 \)	\(=\)	\(1250613\)	\(=\)	\( 3^{3} \cdot 7 \cdot 13 \cdot 509 \)
\( I_{10} \)	\(=\)	\(1752192\)	\(=\)	\( 2^{7} \cdot 3^{4} \cdot 13^{2} \)

Automorphism group

\(\mathrm{Aut}(X)\)	\(\simeq\)	$C_6$
\(\mathrm{Aut}(X_{\overline{\Q}})\)	\(\simeq\)	$D_6$

Rational points

Known points
\((1 : 0 : 0)\)	\((1 : -1 : 0)\)	\((0 : 0 : 1)\)	\((-1 : 0 : 1)\)	\((0 : -1 : 1)\)	\((-1 : 1 : 1)\)
\((2 : -3 : 1)\)	\((-1 : -6 : 3)\)	\((2 : -8 : 1)\)	\((-1 : -11 : 3)\)	\((-3 : 13 : 2)\)	\((-3 : 18 : 2)\)

Number of rational Weierstrass points: \(0\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

Group structure: \(\Z \oplus \Z\)

Generator	$D_0$						Height	Order
\((-1 : 0 : 1) + (0 : 0 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\)	\(x (x + z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(0\)	\(0.150903\)	\(\infty\)
\((0 : -1 : 1) - (1 : -1 : 0)\)	\(z x\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-z^3\)	\(0.150903\)	\(\infty\)

2-torsion field: 9.9.2565164201769.1

BSD invariants

Hasse-Weil conjecture:	verified
Analytic rank:	\(2\)
Mordell-Weil rank:	\(2\)
2-Selmer rank:	\(2\)
Regulator:	\( 0.017078 \)
Real period:	\( 22.36905 \)
Tamagawa product:	\( 1 \)
Torsion order:	\( 1 \)
Leading coefficient:	\( 0.382038 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(3\)	\(4\)	\(4\)	\(1\)	\(1 + 3 T + 3 T^{2}\)
\(13\)	\(2\)	\(2\)	\(1\)	\(1 + 5 T + 13 T^{2}\)

Galois representations

For primes $\ell \ge 5$ the Galois representation data has not been computed for this curve since it is not generic.

For primes $\ell \le 3$, the image of the mod-$\ell$ Galois representation is listed in the table below, whenever it is not all of $\GSp(4,\F_\ell)$.

Prime \(\ell\)	mod-\(\ell\) image	Is torsion prime?
\(2\)	2.80.1	no
\(3\)	3.480.12	no

Sato-Tate group

\(\mathrm{ST}\)	\(\simeq\)	$E_6$
\(\mathrm{ST}^0\)	\(\simeq\)	\(\mathrm{SU}(2)\)

Decomposition of the Jacobian

Splits over the number field \(\Q (b) \simeq \) 6.6.2436053373.2 with defining polynomial:
  \(x^{6} - 39 x^{4} - 26 x^{3} + 351 x^{2} + 585 x + 117\)

Decomposes up to isogeny as the square of the elliptic curve isogeny class:
  \(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
  \(g_4 = -\frac{12015765}{3956} b^{5} + \frac{233936235}{15824} b^{4} + \frac{735217587}{15824} b^{3} - \frac{2326501125}{15824} b^{2} - \frac{240909201}{688} b - \frac{288784197}{3956}\)
  \(g_6 = \frac{36685451205}{3956} b^{5} - \frac{2858709285549}{63296} b^{4} - \frac{8968980313215}{63296} b^{3} + \frac{28420667571231}{63296} b^{2} + \frac{2939506491519}{2752} b + \frac{14100757396245}{63296}\)
   Conductor norm: 1

Endomorphisms of the Jacobian

Of \(\GL_2\)-type over \(\Q\)

Endomorphism ring over \(\Q\):

\(\End (J_{})\)	\(\simeq\)	\(\Z [\frac{1 + \sqrt{-3}}{2}]\)
\(\End (J_{}) \otimes \Q \)	\(\simeq\)	\(\Q(\sqrt{-3}) \)
\(\End (J_{}) \otimes \R\)	\(\simeq\)	\(\C\)

Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) 6.6.2436053373.2 with defining polynomial \(x^{6} - 39 x^{4} - 26 x^{3} + 351 x^{2} + 585 x + 117\)

Not of \(\GL_2\)-type over \(\overline{\Q}\)

Endomorphism ring over \(\overline{\Q}\):

\(\End (J_{\overline{\Q}})\)	\(\simeq\)	an Eichler order of index \(3\) in a maximal order of \(\End (J_{\overline{\Q}}) \otimes \Q\)
\(\End (J_{\overline{\Q}}) \otimes \Q \)	\(\simeq\)	\(\mathrm{M}_2(\)\(\Q\)\()\)
\(\End (J_{\overline{\Q}}) \otimes \R\)	\(\simeq\)	\(\mathrm{M}_2 (\R)\)

Remainder of the endomorphism lattice by field

Over subfield \(F \simeq \) \(\Q(\sqrt{13}) \) with generator \(-\frac{9}{989} a^{5} - \frac{12}{989} a^{4} + \frac{335}{989} a^{3} + \frac{351}{989} a^{2} - \frac{117}{43} a - \frac{1930}{989}\) with minimal polynomial \(x^{2} - x - 3\):

\(\End (J_{F})\)	\(\simeq\)	\(\Z [\frac{1 + \sqrt{-3}}{2}]\)
\(\End (J_{F}) \otimes \Q \)	\(\simeq\)	\(\Q(\sqrt{-3}) \)
\(\End (J_{F}) \otimes \R\)	\(\simeq\)	\(\C\)

  Sato Tate group: $E_3$
  Of \(\GL_2\)-type, simple

Over subfield \(F \simeq \) 3.3.13689.2 with generator \(\frac{4}{129} a^{5} - \frac{3}{43} a^{4} - \frac{56}{43} a^{3} + \frac{145}{129} a^{2} + \frac{585}{43} a + \frac{442}{43}\) with minimal polynomial \(x^{3} - 39 x - 91\):

\(\End (J_{F})\)	\(\simeq\)	\(\Z [\frac{1 + \sqrt{-3}}{2}]\)
\(\End (J_{F}) \otimes \Q \)	\(\simeq\)	\(\Q(\sqrt{-3}) \)
\(\End (J_{F}) \otimes \R\)	\(\simeq\)	\(\C\)

  Sato Tate group: $E_2$
  Of \(\GL_2\)-type, simple