Genus 2 curve 135424.l.270848.1

• Label: 135424.l.270848.1
• Conductor: $135424$
• Discriminant: $-270848$
• Mordell-Weil group: \(\Z/{2}\Z\)
• Sato-Tate group: $E_4$
• \(\End(J_{\overline{\Q}}) \otimes \R\): \(\mathrm{M}_2(\R)\)
• \(\End(J_{\overline{\Q}}) \otimes \Q\): \(\mathrm{M}_2(\Q)\)
• \(\End(J) \otimes \Q\): \(\mathsf{CM}\)
• \(\overline{\Q}\)-simple: no
• \(\mathrm{GL}_2\)-type: yes

Minimal equation

$y^2 + (x^3 + x^2 + x + 1)y = -x^6 - 2x^4 - x^3 - 2x^2 - x - 1$

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(135424\)	\(=\)	\( 2^{8} \cdot 23^{2} \)
Discriminant:	\( \Delta \)	\(=\)	\(-270848\)	\(=\)	\( - 2^{9} \cdot 23^{2} \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(170\)	\(=\)	\( 2 \cdot 5 \cdot 17 \)
\( I_4 \)	\(=\)	\(430\)	\(=\)	\( 2 \cdot 5 \cdot 43 \)
\( I_6 \)	\(=\)	\(23560\)	\(=\)	\( 2^{3} \cdot 5 \cdot 19 \cdot 31 \)
\( I_{10} \)	\(=\)	\(1058\)	\(=\)	\( 2 \cdot 23^{2} \)

Automorphism group

\(\mathrm{Aut}(X)\)	\(\simeq\)	$C_4$
\(\mathrm{Aut}(X_{\overline{\Q}})\)	\(\simeq\)	$D_4$

Rational points

This curve has no rational points.

Number of rational Weierstrass points: \(0\)

This curve is locally solvable except over $\R$.

Mordell-Weil group of the Jacobian

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

Generator	$D_0$						Height	Order
\(D_0 - D_\infty\)	\(x^2 + z^2\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(0\)	\(0\)	\(2\)

2-torsion field: 8.0.34668544.1

BSD invariants

Hasse-Weil conjecture:	verified
Analytic rank:	\(0\)
Mordell-Weil rank:	\(0\)
2-Selmer rank:	\(2\)
Regulator:	\( 1 \)
Real period:	\( 5.075262 \)
Tamagawa product:	\( 2 \)
Torsion order:	\( 2 \)
Leading coefficient:	\( 5.075262 \)
Analytic order of Ш:	\( 2 \)   (rounded)
Order of Ш:	twice a square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(2\)	\(8\)	\(9\)	\(2\)	\(1 - 2 T + 2 T^{2}\)
\(23\)	\(2\)	\(2\)	\(1\)	\(1 + T^{2}\)

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?
\(2\)	2.45.1	yes
\(3\)	3.540.6	no

Sato-Tate group

\(\mathrm{ST}\)	\(\simeq\)	$E_4$
\(\mathrm{ST}^0\)	\(\simeq\)	\(\mathrm{SU}(2)\)

Decomposition of the Jacobian

Splits over the number field \(\Q (b) \simeq \) \(\Q(\zeta_{16})^+\) with defining polynomial:
  \(x^{4} - 4 x^{2} + 2\)

Decomposes up to isogeny as the square of the elliptic curve isogeny class:
  \(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
  \(g_4 = 1600 b^{3} - 2880 b^{2} - 1280 b + 2080\)
  \(g_6 = -484096 b^{3} + 897536 b^{2} + 274688 b - 517632\)
   Conductor norm: 279841

Endomorphisms of the Jacobian

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

Endomorphism ring over \(\Q\):

\(\End (J_{})\)	\(\simeq\)	\(\Z [\sqrt{-1}]\)
\(\End (J_{}) \otimes \Q \)	\(\simeq\)	\(\Q(\sqrt{-1}) \)
\(\End (J_{}) \otimes \R\)	\(\simeq\)	\(\C\)

Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) \(\Q(\zeta_{16})^+\) with defining polynomial \(x^{4} - 4 x^{2} + 2\)

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 \(\End (J_{\overline{\Q}}) \otimes \Q\)
\(\End (J_{\overline{\Q}}) \otimes \Q \)	\(\simeq\)	\(\mathrm{M}_2(\)\(\Q\)\()\)
\(\End (J_{\overline{\Q}}) \otimes \R\)	\(\simeq\)	\(\mathrm{M}_2 (\R)\)

Remainder of the endomorphism lattice by field

Over subfield \(F \simeq \) \(\Q(\sqrt{2}) \) with generator \(a^{2} - 2\) with minimal polynomial \(x^{2} - 2\):

\(\End (J_{F})\)	\(\simeq\)	\(\Z [\sqrt{-1}]\)
\(\End (J_{F}) \otimes \Q \)	\(\simeq\)	\(\Q(\sqrt{-1}) \)
\(\End (J_{F}) \otimes \R\)	\(\simeq\)	\(\C\)

  Sato Tate group: $E_2$
  Of \(\GL_2\)-type, simple