Genus 2 curve 630.a.34020.1

• Label: 630.a.34020.1
• Conductor: $630$
• Discriminant: $34020$
• Mordell-Weil group: $\Z/{2}\Z \oplus \Z/{2}\Z \oplus \Z/{4}\Z$
• 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

Minimal equation

$y^2 + (x^2 + x)y = 3x^5 + 10x^4 - 23x^2 - 6x + 15$

(, )

Invariants

Conductor:	$N$	$=$	$630$	$=$	$2 \cdot 3^{2} \cdot 5 \cdot 7$
Discriminant:	$\Delta$	$=$	$34020$	$=$	$2^{2} \cdot 3^{5} \cdot 5 \cdot 7$

Igusa-Clebsch invariants

$I_2$	$=$	$24100$	$=$	$2^{2} \cdot 5^{2} \cdot 241$
$I_4$	$=$	$969793$	$=$	$11 \cdot 131 \cdot 673$
$I_6$	$=$	$7474503265$	$=$	$5 \cdot 1494900653$
$I_{10}$	$=$	$4354560$	$=$	$2^{9} \cdot 3^{5} \cdot 5 \cdot 7$

Automorphism group

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

Rational points

All points: $(1 : 0 : 0),\, (1 : -1 : 1),\, (-2 : -1 : 1),\, (-5 : -15 : 3)$

Number of rational Weierstrass points: $4$

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

Group structure: $\Z/{2}\Z \oplus \Z/{2}\Z \oplus \Z/{4}\Z$

Generator	$D_0$						Height	Order
$(-2 : -1 : 1) + (1 : -1 : 1) - 2 \cdot(1 : 0 : 0)$	$(x - z) (x + 2z)$	$=$	$0,$	$y$	$=$	$-z^3$	$0$	$2$
$(-2 : -1 : 1) - (1 : 0 : 0)$	$x + 2z$	$=$	$0,$	$y$	$=$	$-z^3$	$0$	$2$
$D_0 - 2 \cdot(1 : 0 : 0)$	$3x^2 + xz - 7z^2$	$=$	$0,$	$y$	$=$	$xz^2 + z^3$	$0$	$4$

2-torsion field: $\Q(\sqrt{105})$

BSD invariants

Hasse-Weil conjecture:	verified
Analytic rank:	$0$
Mordell-Weil rank:	$0$
2-Selmer rank:	$3$
Regulator:	$1$
Real period:	$19.47088$
Tamagawa product:	$4$
Torsion order:	$16$
Leading coefficient:	$0.304232$
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?
$2$	$1$	$2$	$2$	$-1^*$	$( 1 - T )( 1 + T + 2 T^{2} )$		yes
$3$	$2$	$5$	$2$	$1$	$( 1 + T )^{2}$		yes
$5$	$1$	$1$	$1$	$-1$	$( 1 - T )( 1 + 2 T + 5 T^{2} )$		yes
$7$	$1$	$1$	$1$	$1$	$( 1 + T )( 1 + 7 T^{2} )$		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?
$2$	2.360.2	yes
$3$	3.90.1	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

Splits over $\Q$

Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
  Elliptic curve isogeny class 15.a
  Elliptic curve isogeny class 42.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 $\Q$.