Genus 2 curve 20736.l.373248.1

• Label: 20736.l.373248.1
• Conductor: $20736$
• Discriminant: $373248$
• Mordell-Weil group: \(\Z/{2}\Z \oplus \Z/{6}\Z\)
• Sato-Tate group: $J(E_4)$
• \(\End(J_{\overline{\Q}}) \otimes \R\): \(\mathrm{M}_2(\R)\)
• \(\End(J_{\overline{\Q}}) \otimes \Q\): \(\mathsf{QM}\)
• \(\End(J) \otimes \Q\): \(\Q\)
• \(\overline{\Q}\)-simple: yes
• \(\mathrm{GL}_2\)-type: no

Minimal equation

$y^2 + y = 6x^5 + 9x^4 - x^3 - 3x^2$

(, )

Invariants

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

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(146\)	\(=\)	\( 2 \cdot 73 \)
\( I_4 \)	\(=\)	\(738\)	\(=\)	\( 2 \cdot 3^{2} \cdot 41 \)
\( I_6 \)	\(=\)	\(29472\)	\(=\)	\( 2^{5} \cdot 3 \cdot 307 \)
\( I_{10} \)	\(=\)	\(6\)	\(=\)	\( 2 \cdot 3 \)

Automorphism group

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

Rational points

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

Number of rational Weierstrass points: \(2\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

Group structure: \(\Z/{2}\Z \oplus \Z/{6}\Z\)

Generator	$D_0$						Height	Order
\((-1 : -4 : 2) - (1 : 0 : 0)\)	\(2x + z\)	\(=\)	\(0,\)	\(2y\)	\(=\)	\(-z^3\)	\(0\)	\(2\)
\((0 : -1 : 1) - (1 : 0 : 0)\)	\(x\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-z^3\)	\(0\)	\(6\)

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

BSD invariants

Hasse-Weil conjecture:	unverified
Analytic rank:	\(0\)
Mordell-Weil rank:	\(0\)
2-Selmer rank:	\(2\)
Regulator:	\( 1 \)
Real period:	\( 18.60015 \)
Tamagawa product:	\( 8 \)
Torsion order:	\( 12 \)
Leading coefficient:	\( 1.033342 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

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

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?
\(2\)	2.180.3	yes
\(3\)	3.480.3	yes

Sato-Tate group

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

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 \(K = \Q (a) \simeq \) 8.0.339738624.10 with defining polynomial \(x^{8} + 4 x^{6} + 10 x^{4} + 24 x^{2} + 36\)

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

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

\(\End (J_{\overline{\Q}})\)	\(\simeq\)	a non-Eichler order of index \(6\) in a maximal order of \(\End (J_{\overline{\Q}}) \otimes \Q\)
\(\End (J_{\overline{\Q}}) \otimes \Q \)	\(\simeq\)	the quaternion algebra over \(\Q\) of discriminant 6
\(\End (J_{\overline{\Q}}) \otimes \R\)	\(\simeq\)	\(\mathrm{M}_2 (\R)\)

Remainder of the endomorphism lattice by field

Over subfield \(F \simeq \) \(\Q(\sqrt{-3}) \) with generator \(-\frac{1}{15} a^{6} - \frac{1}{6} a^{4} - \frac{2}{3} a^{2} - \frac{3}{5}\) with minimal polynomial \(x^{2} - x + 1\):

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

  Sato Tate group: $E_4$
  Of \(\GL_2\)-type, simple

Over subfield \(F \simeq \) \(\Q(\sqrt{-2}) \) with generator \(-\frac{1}{10} a^{6} - \frac{2}{5}\) with minimal polynomial \(x^{2} + 2\):

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

  Sato Tate group: $J(E_2)$
  Not of \(\GL_2\)-type, simple

Over subfield \(F \simeq \) \(\Q(\sqrt{6}) \) with generator \(-\frac{1}{6} a^{6} - \frac{2}{3} a^{4} - \frac{2}{3} a^{2} - 2\) with minimal polynomial \(x^{2} - 6\):

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

  Sato Tate group: $J(E_2)$
  Not of \(\GL_2\)-type, simple

Over subfield \(F \simeq \) \(\Q(\sqrt{-2}, \sqrt{-3})\) with generator \(-\frac{2}{15} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2} - \frac{6}{5}\) with minimal polynomial \(x^{4} - 2 x^{2} + 4\):

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

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

Over subfield \(F \simeq \) 4.0.6144.1 with generator \(\frac{2}{45} a^{7} - \frac{1}{18} a^{5} + \frac{1}{9} a^{3} + \frac{11}{15} a\) with minimal polynomial \(x^{4} - 4 x^{2} + 6\):

\(\End (J_{F})\)	\(\simeq\)	\(\Z [\sqrt{6}]\)
\(\End (J_{F}) \otimes \Q \)	\(\simeq\)	\(\Q(\sqrt{6}) \)
\(\End (J_{F}) \otimes \R\)	\(\simeq\)	\(\R \times \R\)

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

Over subfield \(F \simeq \) 4.2.18432.2 with generator \(\frac{7}{90} a^{7} + \frac{5}{18} a^{5} + \frac{4}{9} a^{3} + \frac{23}{15} a\) with minimal polynomial \(x^{4} + 4 x^{2} - 2\):

\(\End (J_{F})\)	\(\simeq\)	\(\Z [\sqrt{3}]\)
\(\End (J_{F}) \otimes \Q \)	\(\simeq\)	\(\Q(\sqrt{3}) \)
\(\End (J_{F}) \otimes \R\)	\(\simeq\)	\(\R \times \R\)

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

Over subfield \(F \simeq \) 4.2.18432.2 with generator \(\frac{1}{18} a^{7} + \frac{1}{18} a^{5} - \frac{1}{9} a^{3} - \frac{1}{3} a\) with minimal polynomial \(x^{4} + 4 x^{2} - 2\):

\(\End (J_{F})\)	\(\simeq\)	\(\Z [\sqrt{3}]\)
\(\End (J_{F}) \otimes \Q \)	\(\simeq\)	\(\Q(\sqrt{3}) \)
\(\End (J_{F}) \otimes \R\)	\(\simeq\)	\(\R \times \R\)

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

Over subfield \(F \simeq \) 4.0.6144.1 with generator \(-\frac{1}{15} a^{7} - \frac{1}{6} a^{5} - \frac{2}{3} a^{3} - \frac{3}{5} a\) with minimal polynomial \(x^{4} - 4 x^{2} + 6\):

\(\End (J_{F})\)	\(\simeq\)	\(\Z [\sqrt{6}]\)
\(\End (J_{F}) \otimes \Q \)	\(\simeq\)	\(\Q(\sqrt{6}) \)
\(\End (J_{F}) \otimes \R\)	\(\simeq\)	\(\R \times \R\)

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

Additional information

This curve has the smallest discriminant known for a genus 2 curve with potential QM (the geometric endomorphism algebra of its Jacobian is a non-split quaternion algebra).