Genus 2 curve 45696.b.319872.1

• Label: 45696.b.319872.1
• Conductor: $45696$
• Discriminant: $-319872$
• Mordell-Weil group: \(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
• Sato-Tate group: $\mathrm{SU}(2)\times\mathrm{SU}(2)$
• \(\End(J_{\overline{\Q}}) \otimes \R\): \(\R \times \R\)
• \(\End(J_{\overline{\Q}}) \otimes \Q\): \(\Q \times \Q\)
• \(\End(J) \otimes \Q\): \(\Q \times \Q\)
• \(\overline{\Q}\)-simple: no
• \(\mathrm{GL}_2\)-type: yes

Minimal equation

$y^2 + xy = 3x^6 + 74x^4 + 608x^2 + 1666$

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(45696\)	\(=\)	\( 2^{7} \cdot 3 \cdot 7 \cdot 17 \)
Discriminant:	\( \Delta \)	\(=\)	\(-319872\)	\(=\)	\( - 2^{7} \cdot 3 \cdot 7^{2} \cdot 17 \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(3839376\)	\(=\)	\( 2^{4} \cdot 3 \cdot 79987 \)
\( I_4 \)	\(=\)	\(51981\)	\(=\)	\( 3 \cdot 17327 \)
\( I_6 \)	\(=\)	\(66523796358\)	\(=\)	\( 2 \cdot 3 \cdot 457 \cdot 24261049 \)
\( I_{10} \)	\(=\)	\(39984\)	\(=\)	\( 2^{4} \cdot 3 \cdot 7^{2} \cdot 17 \)

Automorphism group

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

Rational points

This curve has no rational points.

Number of rational Weierstrass points: \(0\)

This curve is locally solvable except over $\Q_{2}$.

Mordell-Weil group of the Jacobian

Group structure: \(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)

Generator	$D_0$						Height	Order
\(D_0 - D_\infty\)	\(3x^2 + 25z^2\)	\(=\)	\(0,\)	\(6y\)	\(=\)	\(-3xz^2 + z^3\)	\(1.351603\)	\(\infty\)
\(D_0 - D_\infty\)	\(2x^2 + 17z^2\)	\(=\)	\(0,\)	\(2y\)	\(=\)	\(-xz^2\)	\(0\)	\(2\)
\(D_0 - D_\infty\)	\(x^2 + 8z^2\)	\(=\)	\(0,\)	\(2y\)	\(=\)	\(-xz^2\)	\(0\)	\(2\)

2-torsion field: 8.0.443364212736.4

BSD invariants

Hasse-Weil conjecture:	verified
Analytic rank:	\(1\)
Mordell-Weil rank:	\(1\)
2-Selmer rank:	\(5\)
Regulator:	\( 1.351603 \)
Real period:	\( 3.298910 \)
Tamagawa product:	\( 2 \)
Torsion order:	\( 4 \)
Leading coefficient:	\( 2.229409 \)
Analytic order of Ш:	\( 4 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(2\)	\(7\)	\(7\)	\(1\)	\(1 - T\)
\(3\)	\(1\)	\(1\)	\(1\)	\(( 1 + T )( 1 + 3 T^{2} )\)
\(7\)	\(1\)	\(2\)	\(2\)	\(( 1 - T )( 1 + 4 T + 7 T^{2} )\)
\(17\)	\(1\)	\(1\)	\(1\)	\(( 1 - T )( 1 + 6 T + 17 T^{2} )\)

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?
\(2\)	2.90.6	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 192.a
  Elliptic curve isogeny class 238.d

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 \(\Q\).