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⁡(JQ‾)⊗R\End(J_{\overline{\Q}}) \otimes \R: $\mathrm{M}_2(\R)$
• End⁡(JQ‾)⊗Q\End(J_{\overline{\Q}}) \otimes \Q: $\mathrm{M}_2(\Q)$
• End⁡(J)⊗Q\End(J) \otimes \Q: $\mathsf{CM}$
• Q‾\overline{\Q}-simple: no
• GL2\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	Root number*	L-factor	Cluster picture	Tame reduction?
$3$	$4$	$4$	$1$	$1^*$	$1 + 3 T + 3 T^{2}$		no
$13$	$2$	$2$	$1$	$1$	$1 + 5 T + 13 T^{2}$		yes

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