Genus 2 curve 692224.a.692224.1

• Label: 692224.a.692224.1
• Conductor: $692224$
• Discriminant: $-692224$
• Mordell-Weil group: \(\Z/{2}\Z\)
• Sato-Tate group: $J(E_2)$
• \(\End(J_{\overline{\Q}}) \otimes \R\): \(\mathrm{M}_2(\R)\)
• \(\End(J_{\overline{\Q}}) \otimes \Q\): \(\mathrm{M}_2(\Q)\)
• \(\End(J) \otimes \Q\): \(\Q\)
• \(\overline{\Q}\)-simple: no
• \(\mathrm{GL}_2\)-type: no

Minimal equation

$y^2 = x^5 - 3x^3 - x$

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(692224\)	\(=\)	\( 2^{12} \cdot 13^{2} \)
Discriminant:	\( \Delta \)	\(=\)	\(-692224\)	\(=\)	\( - 2^{12} \cdot 13^{2} \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(14\)	\(=\)	\( 2 \cdot 7 \)
\( I_4 \)	\(=\)	\(-404\)	\(=\)	\( - 2^{2} \cdot 101 \)
\( I_6 \)	\(=\)	\(-3928\)	\(=\)	\( - 2^{3} \cdot 491 \)
\( I_{10} \)	\(=\)	\(-2704\)	\(=\)	\( - 2^{4} \cdot 13^{2} \)

Automorphism group

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

Rational points

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

Number of rational Weierstrass points: \(2\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

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

Generator	$D_0$						Height	Order
\((0 : 0 : 1) - (1 : 0 : 0)\)	\(x\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(0\)	\(0\)	\(2\)

2-torsion field: 4.0.832.1

BSD invariants

Hasse-Weil conjecture:	unverified
Analytic rank:	\(0\)
Mordell-Weil rank:	\(0\)
2-Selmer rank:	\(1\)
Regulator:	\( 1 \)
Real period:	\( 9.748987 \)
Tamagawa product:	\( 1 \)
Torsion order:	\( 2 \)
Leading coefficient:	\( 2.437246 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(2\)	\(12\)	\(12\)	\(1\)	\(1\)
\(13\)	\(2\)	\(2\)	\(1\)	\(1 + T^{2}\)

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?
\(2\)	2.90.3	yes
\(3\)	3.540.2	no

Sato-Tate group

\(\mathrm{ST}\)	\(\simeq\)	$J(E_2)$
\(\mathrm{ST}^0\)	\(\simeq\)	\(\mathrm{SU}(2)\)

Decomposition of the Jacobian

Splits over the number field \(\Q (b) \simeq \) \(\Q(\zeta_{8})\) with defining polynomial:
  \(x^{4} + 1\)

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
  \(g_4 = -\frac{28160}{28561} b^{2} + \frac{581376}{28561}\)
  \(g_6 = -\frac{416530432}{4826809} b^{3} + \frac{355147776}{4826809} b\)
   Conductor norm: 7311616

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 \(K = \Q (a) \simeq \) \(\Q(\zeta_{8})\) with defining polynomial \(x^{4} + 1\)

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

Remainder of the endomorphism lattice by field

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

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

  Sato Tate group: $J(E_1)$
  Of \(\GL_2\)-type, simple

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

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

  Sato Tate group: $J(E_1)$
  Of \(\GL_2\)-type, simple

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

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

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