Genus 2 curve 1152.a.147456.1

• Label: 1152.a.147456.1
• Conductor: $1152$
• Discriminant: $147456$
• Mordell-Weil group: \(\Z/{8}\Z\)
• Sato-Tate group: $J(E_1)$
• \(\End(J_{\overline{\Q}}) \otimes \R\): \(\mathrm{M}_2(\R)\)
• \(\End(J_{\overline{\Q}}) \otimes \Q\): \(\mathrm{M}_2(\Q)\)
• \(\End(J) \otimes \Q\): \(\Q \times \Q\)
• \(\overline{\Q}\)-simple: no
• \(\mathrm{GL}_2\)-type: yes

Minimal equation

$y^2 = x^6 - 2x^4 + 2x^2 - 1$

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(1152\)	\(=\)	\( 2^{7} \cdot 3^{2} \)
Discriminant:	\( \Delta \)	\(=\)	\(147456\)	\(=\)	\( 2^{14} \cdot 3^{2} \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(152\)	\(=\)	\( 2^{3} \cdot 19 \)
\( I_4 \)	\(=\)	\(109\)	\(=\)	\( 109 \)
\( I_6 \)	\(=\)	\(5469\)	\(=\)	\( 3 \cdot 1823 \)
\( I_{10} \)	\(=\)	\(18\)	\(=\)	\( 2 \cdot 3^{2} \)

Automorphism group

\(\mathrm{Aut}(X)\)	\(\simeq\)	$C_2^2$
\(\mathrm{Aut}(X_{\overline{\Q}})\)	\(\simeq\)	$D_4$

Rational points

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

Number of rational Weierstrass points: \(2\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

Group structure: \(\Z/{8}\Z\)

Generator	$D_0$						Height	Order
\((-1 : 0 : 1) - (1 : -1 : 0)\)	\(z (x + z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(x^3 + z^3\)	\(0\)	\(8\)

2-torsion field: \(\Q(\zeta_{12})\)

BSD invariants

Hasse-Weil conjecture:	verified
Analytic rank:	\(0\)
Mordell-Weil rank:	\(0\)
2-Selmer rank:	\(1\)
Regulator:	\( 1 \)
Real period:	\( 7.270694 \)
Tamagawa product:	\( 4 \)
Torsion order:	\( 8 \)
Leading coefficient:	\( 0.454418 \)
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?
\(2\)	\(7\)	\(14\)	\(4\)	\(-1^*\)	\(1\)		no
\(3\)	\(2\)	\(2\)	\(1\)	\(-1\)	\(( 1 - T )( 1 + T )\)		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?
\(2\)	2.180.5	yes
\(3\)	3.1080.10	no

Sato-Tate group

\(\mathrm{ST}\)	\(\simeq\)	$J(E_1)$
\(\mathrm{ST}^0\)	\(\simeq\)	\(\mathrm{SU}(2)\)

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 24.a
  Elliptic curve isogeny class 48.a

Endomorphisms of the Jacobian

Of \(\GL_2\)-type over \(\Q\)

Endomorphism ring over \(\Q\):

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

Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) \(\Q(\sqrt{-1}) \) with defining polynomial \(x^{2} + 1\)

Not of \(\GL_2\)-type over \(\overline{\Q}\)

Endomorphism ring over \(\overline{\Q}\):

\(\End (J_{\overline{\Q}})\)	\(\simeq\)	a non-Eichler order of index \(4\) in a maximal order of \(\End (J_{\overline{\Q}}) \otimes \Q\)
\(\End (J_{\overline{\Q}}) \otimes \Q \)	\(\simeq\)	\(\mathrm{M}_2(\)\(\Q\)\()\)
\(\End (J_{\overline{\Q}}) \otimes \R\)	\(\simeq\)	\(\mathrm{M}_2 (\R)\)