Genus 2 curve 1369.a.50653.1

• Label: 1369.a.50653.1
• Conductor: $1369$
• Discriminant: $50653$
• Mordell-Weil group: $\Z \oplus \Z/{3}\Z$
• 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: $\Q \times \Q$
• End⁡(J)⊗Q\End(J) \otimes \Q: $\Q \times \Q$
• Q‾\overline{\Q}-simple: no
• GL2\mathrm{GL}_2-type: yes

This is a model for the modular curve $X_0(37)$.

Minimal equation

$y^2 + x^3y = 2x^5 - 5x^4 + 7x^3 - 6x^2 + 3x - 1$

(, )

Invariants

Conductor:	$N$	$=$	$1369$	$=$	$37^{2}$
Discriminant:	$\Delta$	$=$	$50653$	$=$	$37^{3}$

Igusa-Clebsch invariants

$I_2$	$=$	$456$	$=$	$2^{3} \cdot 3 \cdot 19$
$I_4$	$=$	$11220$	$=$	$2^{2} \cdot 3 \cdot 5 \cdot 11 \cdot 17$
$I_6$	$=$	$2199936$	$=$	$2^{7} \cdot 3 \cdot 17 \cdot 337$
$I_{10}$	$=$	$202612$	$=$	$2^{2} \cdot 37^{3}$

Automorphism group

$\mathrm{Aut}(X)$	$\simeq$	$C_2^2$
$\mathrm{Aut}(X_{\overline{\Q}})$	$\simeq$	$C_2^2$

Rational points

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

Number of rational Weierstrass points: $0$

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

Group structure: $\Z \oplus \Z/{3}\Z$

Generator	$D_0$						Height	Order
$D_0 - (1 : -1 : 0) - (1 : 0 : 0)$	$x^2 - xz + z^2$	$=$	$0,$	$y$	$=$	$z^3$	$0.102222$	$\infty$
$(1 : -1 : 1) - (1 : -1 : 0)$	$z (x - z)$	$=$	$0,$	$y$	$=$	$-z^3$	$0$	$3$

2-torsion field: 4.0.592.1

BSD invariants

Hasse-Weil conjecture:	verified
Analytic rank:	$1$
Mordell-Weil rank:	$1$
2-Selmer rank:	$1$
Regulator:	$0.102222$
Real period:	$6.516888$
Tamagawa product:	$3$
Torsion order:	$3$
Leading coefficient:	$0.222058$
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?
$37$	$2$	$3$	$3$	$-1$	$( 1 - T )( 1 + T )$		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?
$2$	2.30.4	no
$3$	3.2160.20	yes

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

Splits over $\Q$

Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
  Elliptic curve isogeny class 37.b
  Elliptic curve isogeny class 37.a

Endomorphisms of the Jacobian

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

Endomorphism ring over $\Q$:

$\End (J_{})$	$\simeq$	an order of index $2$ in $\Z \times \Z$
$\End (J_{}) \otimes \Q$	$\simeq$	$\Q$ $\times$ $\Q$
$\End (J_{}) \otimes \R$	$\simeq$	$\R \times \R$

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