Genus 2 curve 26244.d.314928.1

• Label: 26244.d.314928.1
• Conductor: $26244$
• Discriminant: $314928$
• Mordell-Weil group: \(\Z \oplus \Z/{3}\Z \oplus \Z/{3}\Z\)
• Sato-Tate group: $J(E_3)$
• \(\End(J_{\overline{\Q}}) \otimes \R\): \(\mathrm{M}_2(\R)\)
• \(\End(J_{\overline{\Q}}) \otimes \Q\): \(\mathrm{M}_2(\Q)\)
• \(\End(J) \otimes \Q\): \(\Q\)
• \(\overline{\Q}\)-simple: no
• \(\mathrm{GL}_2\)-type: no

Minimal equation

$y^2 + y = x^6 - 2x^3$

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(26244\)	\(=\)	\( 2^{2} \cdot 3^{8} \)
Discriminant:	\( \Delta \)	\(=\)	\(314928\)	\(=\)	\( 2^{4} \cdot 3^{9} \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(24\)	\(=\)	\( 2^{3} \cdot 3 \)
\( I_4 \)	\(=\)	\(189\)	\(=\)	\( 3^{3} \cdot 7 \)
\( I_6 \)	\(=\)	\(1107\)	\(=\)	\( 3^{3} \cdot 41 \)
\( I_{10} \)	\(=\)	\(-162\)	\(=\)	\( - 2 \cdot 3^{4} \)

Automorphism group

\(\mathrm{Aut}(X)\)	\(\simeq\)	$C_2$
\(\mathrm{Aut}(X_{\overline{\Q}})\)	\(\simeq\)	$D_6$

Rational points

All points: \((1 : -1 : 0),\, (1 : 1 : 0),\, (0 : 0 : 1),\, (0 : -1 : 1),\, (1 : -3 : 2),\, (1 : -5 : 2)\)

Number of rational Weierstrass points: \(0\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

Group structure: \(\Z \oplus \Z/{3}\Z \oplus \Z/{3}\Z\)

Generator	$D_0$						Height	Order
\((1 : -5 : 2) - (1 : -1 : 0)\)	\(z (2x - z)\)	\(=\)	\(0,\)	\(4y\)	\(=\)	\(4x^3 - 3z^3\)	\(0.985564\)	\(\infty\)
\((0 : -1 : 1) - (1 : -1 : 0)\)	\(z x\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(x^3 - z^3\)	\(0\)	\(3\)
\(D_0 - 2 \cdot(1 : 1 : 0)\)	\(z^2\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-x^3\)	\(0\)	\(3\)

2-torsion field: 6.2.5038848.1

BSD invariants

Hasse-Weil conjecture:	unverified
Analytic rank:	\(1\)
Mordell-Weil rank:	\(1\)
2-Selmer rank:	\(1\)
Regulator:	\( 0.985564 \)
Real period:	\( 12.73964 \)
Tamagawa product:	\( 9 \)
Torsion order:	\( 9 \)
Leading coefficient:	\( 1.395082 \)
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?
\(2\)	\(2\)	\(4\)	\(3\)	\(-1^*\)	\(1 + 2 T^{2}\)		yes
\(3\)	\(8\)	\(9\)	\(3\)	\(1^*\)	\(1\)		no

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?
\(2\)	2.20.3	no
\(3\)	3.5760.3	yes

Sato-Tate group

\(\mathrm{ST}\)	\(\simeq\)	$J(E_3)$
\(\mathrm{ST}^0\)	\(\simeq\)	\(\mathrm{SU}(2)\)

Decomposition of the Jacobian

Splits over the number field \(\Q (b) \simeq \) 3.1.108.1 with defining polynomial:
  \(x^{3} - 2\)

Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
  \(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
  \(g_4 = -108 b^{2}\)
  \(g_6 = 3888\)
   Conductor norm: 729
  \(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
  \(g_4 = 468 b^{2}\)
  \(g_6 = -20304\)
   Conductor norm: 729

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 \(K = \Q (a) \simeq \) 6.0.34992.1 with defining polynomial \(x^{6} - 3 x^{5} + 5 x^{3} - 3 x + 1\)

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

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

\(\End (J_{\overline{\Q}})\)	\(\simeq\)	an Eichler order of index \(3\) 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 \(F \simeq \) \(\Q(\sqrt{-3}) \) with generator \(2 a^{5} - 5 a^{4} - 2 a^{3} + 8 a^{2} + 4 a - 3\) with minimal polynomial \(x^{2} - x + 1\):

\(\End (J_{F})\)	\(\simeq\)	\(\Z [\frac{1 + \sqrt{-3}}{2}]\)
\(\End (J_{F}) \otimes \Q \)	\(\simeq\)	\(\Q(\sqrt{-3}) \)
\(\End (J_{F}) \otimes \R\)	\(\simeq\)	\(\C\)

  Sato Tate group: $E_3$
  Of \(\GL_2\)-type, simple

Over subfield \(F \simeq \) 3.1.108.1 with generator \(2 a^{5} - 5 a^{4} - 3 a^{3} + 10 a^{2} + 5 a - 5\) with minimal polynomial \(x^{3} - 2\):

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

  Sato Tate group: $J(E_1)$
  Of \(\GL_2\)-type, not simple

Over subfield \(F \simeq \) 3.1.108.1 with generator \(-a^{2} + a + 1\) with minimal polynomial \(x^{3} - 2\):

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

  Sato Tate group: $J(E_1)$
  Of \(\GL_2\)-type, not simple

Over subfield \(F \simeq \) 3.1.108.1 with generator \(-2 a^{5} + 5 a^{4} + 3 a^{3} - 9 a^{2} - 6 a + 4\) with minimal polynomial \(x^{3} - 2\):

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

  Sato Tate group: $J(E_1)$
  Of \(\GL_2\)-type, not simple