Genus 2 curve 20736.i.373248.1

• Label: 20736.i.373248.1
• Conductor: $20736$
• Discriminant: $373248$
• Mordell-Weil group: \(\Z/{6}\Z\)
• Sato-Tate group: $D_{2,1}$
• \(\End(J_{\overline{\Q}}) \otimes \R\): \(\mathrm{M}_2(\C)\)
• \(\End(J_{\overline{\Q}}) \otimes \Q\): \(\mathrm{M}_2(\mathsf{CM})\)
• \(\End(J) \otimes \Q\): \(\Q \times \Q\)
• \(\overline{\Q}\)-simple: no
• \(\mathrm{GL}_2\)-type: yes

Minimal equation

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

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(20736\)	\(=\)	\( 2^{8} \cdot 3^{4} \)
Discriminant:	\( \Delta \)	\(=\)	\(373248\)	\(=\)	\( 2^{9} \cdot 3^{6} \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(40\)	\(=\)	\( 2^{3} \cdot 5 \)
\( I_4 \)	\(=\)	\(45\)	\(=\)	\( 3^{2} \cdot 5 \)
\( I_6 \)	\(=\)	\(555\)	\(=\)	\( 3 \cdot 5 \cdot 37 \)
\( I_{10} \)	\(=\)	\(6\)	\(=\)	\( 2 \cdot 3 \)

Automorphism group

\(\mathrm{Aut}(X)\)	\(\simeq\)	$C_2^2$
\(\mathrm{Aut}(X_{\overline{\Q}})\)	\(\simeq\)	$C_3:D_4$

Rational points

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

Number of rational Weierstrass points: \(0\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

Group structure: \(\Z/{6}\Z\)

Generator	$D_0$						Height	Order
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\)	\(x^2 + z^2\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(2xz^2\)	\(0\)	\(6\)

2-torsion field: \(\Q(\sqrt{2}, \sqrt{-3})\)

BSD invariants

Hasse-Weil conjecture:	verified
Analytic rank:	\(0\)
Mordell-Weil rank:	\(0\)
2-Selmer rank:	\(1\)
Regulator:	\( 1 \)
Real period:	\( 7.223966 \)
Tamagawa product:	\( 6 \)
Torsion order:	\( 6 \)
Leading coefficient:	\( 1.203994 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(2\)	\(8\)	\(9\)	\(3\)	\(1\)
\(3\)	\(4\)	\(6\)	\(2\)	\(1\)

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?
\(2\)	2.180.4	yes
\(3\)	3.8640.8	yes

Sato-Tate group

\(\mathrm{ST}\)	\(\simeq\)	$D_{2,1}$
\(\mathrm{ST}^0\)	\(\simeq\)	\(\mathrm{U}(1)\)

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 576.f
  Elliptic curve isogeny class 36.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 \(K = \Q (a) \simeq \) \(\Q(\sqrt{-2}, \sqrt{-3})\) with defining polynomial \(x^{4} - 2 x^{2} + 4\)

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

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

\(\End (J_{\overline{\Q}})\)	\(\simeq\)	a non-Eichler order of index \(16\) in a maximal order of \(\End (J_{\overline{\Q}}) \otimes \Q\)
\(\End (J_{\overline{\Q}}) \otimes \Q \)	\(\simeq\)	\(\mathrm{M}_2(\)\(\Q(\sqrt{-3}) \)\()\)
\(\End (J_{\overline{\Q}}) \otimes \R\)	\(\simeq\)	\(\mathrm{M}_2 (\C)\)

Remainder of the endomorphism lattice by field

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

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

  Sato Tate group: $C_{2,1}$
  Not of \(\GL_2\)-type, not simple

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

\(\End (J_{F})\)	\(\simeq\)	an order of index \(4\) in \(\Z [\frac{1 + \sqrt{-3}}{2}] \times \Z [\frac{1 + \sqrt{-3}}{2}]\)
\(\End (J_{F}) \otimes \Q \)	\(\simeq\)	\(\Q(\sqrt{-3}) \) \(\times\) \(\Q(\sqrt{-3}) \)
\(\End (J_{F}) \otimes \R\)	\(\simeq\)	\(\C \times \C\)

  Sato Tate group: $C_2$
  Not of \(\GL_2\)-type, not simple

Over subfield \(F \simeq \) \(\Q(\sqrt{6}) \) with generator \(\frac{1}{2} a^{3} - 2 a\) with minimal polynomial \(x^{2} - 6\):

\(\End (J_{F})\)	\(\simeq\)	a non-Eichler order of index \(12\) in a maximal order of \(\End (J_{F}) \otimes \Q\)
\(\End (J_{F}) \otimes \Q \)	\(\simeq\)	\(\mathrm{M}_2(\)\(\Q\)\()\)
\(\End (J_{F}) \otimes \R\)	\(\simeq\)	\(\mathrm{M}_2 (\R)\)

  Sato Tate group: $C_{2,1}$
  Not of \(\GL_2\)-type, not simple