Genus 2 curve 3969.c.35721.1

• Label: 3969.c.35721.1
• Conductor: $3969$
• Discriminant: $35721$
• Mordell-Weil group: $\Z/{2}\Z \oplus \Z/{2}\Z$
• 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

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

(, )

Invariants

Conductor:	$N$	$=$	$3969$	$=$	$3^{4} \cdot 7^{2}$
Discriminant:	$\Delta$	$=$	$35721$	$=$	$3^{6} \cdot 7^{2}$

Igusa-Clebsch invariants

$I_2$	$=$	$268$	$=$	$2^{2} \cdot 67$
$I_4$	$=$	$2961$	$=$	$3^{2} \cdot 7 \cdot 47$
$I_6$	$=$	$216951$	$=$	$3 \cdot 7 \cdot 10331$
$I_{10}$	$=$	$18816$	$=$	$2^{7} \cdot 3 \cdot 7^{2}$

Automorphism group

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

Rational points

All points: $(1 : 0 : 0),\, (0 : 0 : 1),\, (1 : -1 : 1)$

Number of rational Weierstrass points: $3$

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

Group structure: $\Z/{2}\Z \oplus \Z/{2}\Z$

Generator	$D_0$						Height	Order
$(1 : -1 : 1) - (1 : 0 : 0)$	$x - z$	$=$	$0,$	$y$	$=$	$-z^3$	$0$	$2$
$(0 : 0 : 1) + (1 : -1 : 1) - 2 \cdot(1 : 0 : 0)$	$x (x - z)$	$=$	$0,$	$y$	$=$	$-xz^2$	$0$	$2$

2-torsion field: 3.3.3969.1

BSD invariants

Hasse-Weil conjecture:	verified
Analytic rank:	$0$
Mordell-Weil rank:	$0$
2-Selmer rank:	$2$
Regulator:	$1$
Real period:	$12.48506$
Tamagawa product:	$1$
Torsion order:	$4$
Leading coefficient:	$0.780316$
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?
$3$	$4$	$6$	$1$	$-1^*$	$1 + 3 T^{2}$		no
$7$	$2$	$2$	$1$	$-1$	$1 - 4 T + 7 T^{2}$		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?
$2$	2.240.1	yes
$3$	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 $\Q (b) \simeq$ 6.6.330812181.2 with defining polynomial:
  $x^{6} - 21 x^{4} - 14 x^{3} + 63 x^{2} + 84 x + 28$

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
  $g_4 = -\frac{171315}{2048} b^{5} - \frac{209223}{1024} b^{4} + \frac{2733507}{2048} b^{3} + \frac{1175391}{256} b^{2} + \frac{9537507}{2048} b + \frac{1587033}{1024}$
  $g_6 = \frac{55433889}{4096} b^{5} + \frac{120007251}{4096} b^{4} - \frac{900796869}{4096} b^{3} - \frac{2712953817}{4096} b^{2} - \frac{2417459499}{4096} b - \frac{689012163}{4096}$
   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 $K = \Q (a) \simeq$ 6.6.330812181.2 with defining polynomial $x^{6} - 21 x^{4} - 14 x^{3} + 63 x^{2} + 84 x + 28$

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 $F \simeq$ $\Q(\sqrt{21})$ with generator $-\frac{9}{8} a^{5} + \frac{3}{4} a^{4} + \frac{185}{8} a^{3} - \frac{567}{8} a - \frac{173}{4}$ with minimal polynomial $x^{2} - x - 5$:

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

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

Over subfield $F \simeq$ 3.3.3969.1 with generator $a^{5} - a^{4} - 20 a^{3} + 6 a^{2} + 56 a + 28$ with minimal polynomial $x^{3} - 21 x - 28$:

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

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