Genus 2 curve 28561.a.371293.1

• Label: 28561.a.371293.1
• Conductor: $28561$
• Discriminant: $371293$
• Mordell-Weil group: \(\Z\)
• Sato-Tate group: $F_{ac}$
• \(\End(J_{\overline{\Q}}) \otimes \R\): \(\C \times \C\)
• \(\End(J_{\overline{\Q}}) \otimes \Q\): \(\mathsf{CM}\)
• \(\End(J) \otimes \Q\): \(\Q\)
• \(\overline{\Q}\)-simple: yes
• \(\mathrm{GL}_2\)-type: no

Minimal equation

$y^2 + x^3y = -2x^4 - 2x^3 + 2x^2 + 3x - 2$

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(28561\)	\(=\)	\( 13^{4} \)
Discriminant:	\( \Delta \)	\(=\)	\(371293\)	\(=\)	\( 13^{5} \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(64\)	\(=\)	\( 2^{6} \)
\( I_4 \)	\(=\)	\(-80\)	\(=\)	\( - 2^{4} \cdot 5 \)
\( I_6 \)	\(=\)	\(-1968\)	\(=\)	\( - 2^{4} \cdot 3 \cdot 41 \)
\( I_{10} \)	\(=\)	\(4\)	\(=\)	\( 2^{2} \)

Automorphism group

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

Rational points

All points: \((1 : 0 : 0),\, (1 : -1 : 0),\, (-2 : 4 : 1)\)

Number of rational Weierstrass points: \(1\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

Group structure: \(\Z\)

Generator	$D_0$						Height	Order
\((-2 : 4 : 1) - (1 : 0 : 0)\)	\(z (x + 2z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-x^3 - 4z^3\)	\(0.167945\)	\(\infty\)

2-torsion field: 5.1.35152.1

BSD invariants

Hasse-Weil conjecture:	unverified
Analytic rank:	\(1\)
Mordell-Weil rank:	\(1\)
2-Selmer rank:	\(1\)
Regulator:	\( 0.167945 \)
Real period:	\( 3.378217 \)
Tamagawa product:	\( 2 \)
Torsion order:	\( 1 \)
Leading coefficient:	\( 1.134713 \)
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?
\(13\)	\(4\)	\(5\)	\(2\)	\(-1\)	\(1\)		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?
\(2\)	2.36.1	no
\(3\)	3.3240.17	no

Sato-Tate group

\(\mathrm{ST}\)	\(\simeq\)	$F_{ac}$
\(\mathrm{ST}^0\)	\(\simeq\)	\(\mathrm{U}(1)\times\mathrm{U}(1)\)

Decomposition of the Jacobian

Simple over \(\overline{\Q}\)

Endomorphisms of the Jacobian

Not of \(\GL_2\)-type over \(\Q\)

Endomorphism ring over \(\Q\):

\(\End (J_{})\)	\(\simeq\)	\(\Z\)
\(\End (J_{}) \otimes \Q \)	\(\simeq\)	\(\Q\)
\(\End (J_{}) \otimes \R\)	\(\simeq\)	\(\R\)

Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) \(\Q(\sqrt{-26 -6 \sqrt{13}})\) with defining polynomial \(x^{4} - x^{3} + 2 x^{2} + 4 x + 3\)

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

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

\(\End (J_{\overline{\Q}})\)	\(\simeq\)	the maximal order of \(\End (J_{\overline{\Q}}) \otimes \Q\)
\(\End (J_{\overline{\Q}}) \otimes \Q \)	\(\simeq\)	\(\Q(\sqrt{-26 -6 \sqrt{13}})\) (CM)
\(\End (J_{\overline{\Q}}) \otimes \R\)	\(\simeq\)	\(\C \times \C\)

Remainder of the endomorphism lattice by field

Over subfield \(F \simeq \) \(\Q(\sqrt{13}) \) with generator \(-\frac{1}{3} a^{3} + \frac{1}{3} a - 1\) with minimal polynomial \(x^{2} - x - 3\):

\(\End (J_{F})\)	\(\simeq\)	\(\Z [\frac{1 + \sqrt{13}}{2}]\)
\(\End (J_{F}) \otimes \Q \)	\(\simeq\)	\(\Q(\sqrt{13}) \)
\(\End (J_{F}) \otimes \R\)	\(\simeq\)	\(\R \times \R\)

  Sato Tate group: F_{ab}
  Of \(\GL_2\)-type, simple