Genus 2 curve 841.a.841.1

• Label: 841.a.841.1
• Conductor: $841$
• Discriminant: $-841$
• Mordell-Weil group: $\Z/{7}\Z$
• Sato-Tate group: $\mathrm{SU}(2)\times\mathrm{SU}(2)$
• End⁡(JQ‾)⊗R\End(J_{\overline{\Q}}) \otimes \R: $\R \times \R$
• End⁡(JQ‾)⊗Q\End(J_{\overline{\Q}}) \otimes \Q: $\mathsf{RM}$
• End⁡(J)⊗Q\End(J) \otimes \Q: $\mathsf{RM}$
• Q‾\overline{\Q}-simple: yes
• GL2\mathrm{GL}_2-type: yes

The Jacobian of this curve is isogenous to that of the modular curve $X_0(29)$ (which has discriminant $29^5$).

Minimal equation

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

(, )

Invariants

Conductor:	$N$	$=$	$841$	$=$	$29^{2}$
Discriminant:	$\Delta$	$=$	$-841$	$=$	$- 29^{2}$

Igusa-Clebsch invariants

$I_2$	$=$	$1420$	$=$	$2^{2} \cdot 5 \cdot 71$
$I_4$	$=$	$4201$	$=$	$4201$
$I_6$	$=$	$1973899$	$=$	$61 \cdot 32359$
$I_{10}$	$=$	$107648$	$=$	$2^{7} \cdot 29^{2}$

Automorphism group

$\mathrm{Aut}(X)$	$\simeq$	$C_2$
$\mathrm{Aut}(X_{\overline{\Q}})$	$\simeq$	$C_2$

Rational points

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

Number of rational Weierstrass points: $0$

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

Group structure: $\Z/{7}\Z$

Generator	$D_0$						Height	Order
$D_0 - (1 : -1 : 0) - (1 : 0 : 0)$	$x^2 + z^2$	$=$	$0,$	$y$	$=$	$z^3$	$0$	$7$

2-torsion field: 6.0.53824.1

BSD invariants

Hasse-Weil conjecture:	verified
Analytic rank:	$0$
Mordell-Weil rank:	$0$
2-Selmer rank:	$0$
Regulator:	$1$
Real period:	$14.28455$
Tamagawa product:	$1$
Torsion order:	$7$
Leading coefficient:	$0.291521$
Analytic order of Ш:	$1$   (rounded)
Order of Ш:	square

Local invariants

Prime	ord($N$)	ord($\Delta$)	Tamagawa	L-factor	Cluster picture
$29$	$2$	$2$	$1$	$( 1 - T )^{2}$

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?
$2$	2.60.2	no
$3$	3.72.2	no

Sato-Tate group

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

Decomposition of the Jacobian

Simple over $\overline{\Q}$

Endomorphisms of the Jacobian

Of $\GL_2$-type over $\Q$

Endomorphism ring over $\Q$:

$\End (J_{})$	$\simeq$	$\Z [\sqrt{2}]$
$\End (J_{}) \otimes \Q$	$\simeq$	$\Q(\sqrt{2})$
$\End (J_{}) \otimes \R$	$\simeq$	$\R \times \R$

All $\overline{\Q}$-endomorphisms of the Jacobian are defined over $\Q$.