Genus 2 curve 186624.d.373248.1

• Label: 186624.d.373248.1
• Conductor: $186624$
• Discriminant: $373248$
• Mordell-Weil group: \(\Z \oplus \Z/{3}\Z\)
• 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

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

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(186624\)	\(=\)	\( 2^{8} \cdot 3^{6} \)
Discriminant:	\( \Delta \)	\(=\)	\(373248\)	\(=\)	\( 2^{9} \cdot 3^{6} \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(32\)	\(=\)	\( 2^{5} \)
\( I_4 \)	\(=\)	\(117\)	\(=\)	\( 3^{2} \cdot 13 \)
\( I_6 \)	\(=\)	\(879\)	\(=\)	\( 3 \cdot 293 \)
\( I_{10} \)	\(=\)	\(-6\)	\(=\)	\( - 2 \cdot 3 \)

Automorphism group

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

Rational points

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

Number of rational Weierstrass points: \(0\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

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

Generator	$D_0$						Height	Order
\((-1 : 0 : 1) - (1 : -1 : 0)\)	\(z (x + z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(0\)	\(0.721206\)	\(\infty\)
\(D_0 - 2 \cdot(1 : -1 : 0)\)	\(z^2\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(0\)	\(0\)	\(3\)

2-torsion field: 6.0.139968.1

BSD invariants

Hasse-Weil conjecture:	unverified
Analytic rank:	\(1\)
Mordell-Weil rank:	\(1\)
2-Selmer rank:	\(1\)
Regulator:	\( 0.721206 \)
Real period:	\( 11.73095 \)
Tamagawa product:	\( 3 \)
Torsion order:	\( 3 \)
Leading coefficient:	\( 2.820144 \)
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?
\(2\)	\(8\)	\(9\)	\(3\)	\(-1^*\)	\(1\)		no
\(3\)	\(6\)	\(6\)	\(1\)	\(1^*\)	\(1\)		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?
\(2\)	2.60.2	no
\(3\)	3.2880.5	yes

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 \(\Q (b) \simeq \) \(\Q(\sqrt{2}) \) with defining polynomial:
  \(x^{2} - 2\)

Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
  Elliptic curve isogeny class 2.2.8.1-2916.1-c
  Elliptic curve isogeny class 2.2.8.1-2916.1-b

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 \(K = \Q (a) \simeq \) \(\Q(\sqrt{2}, \sqrt{-3})\) with defining polynomial \(x^{4} + 2 x^{2} + 4\)

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

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

\(\End (J_{\overline{\Q}})\)	\(\simeq\)	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 \(F \simeq \) \(\Q(\sqrt{2}) \) with generator \(\frac{1}{2} a^{3}\) with minimal polynomial \(x^{2} - 2\):

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

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

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

Over subfield \(F \simeq \) \(\Q(\sqrt{-3}) \) with generator \(-\frac{1}{2} a^{2}\) 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: $E_2$
  Of \(\GL_2\)-type, simple