Genus 2 curve 324.a.648.1

• Label: 324.a.648.1
• Conductor: $324$
• Discriminant: $-648$
• Mordell-Weil group: \(\Z/{21}\Z\)
• Sato-Tate group: $E_3$
• \(\End(J_{\overline{\Q}}) \otimes \R\): \(\mathrm{M}_2(\R)\)
• \(\End(J_{\overline{\Q}}) \otimes \Q\): \(\mathrm{M}_2(\Q)\)
• \(\End(J) \otimes \Q\): \(\mathsf{CM}\)
• \(\overline{\Q}\)-simple: no
• \(\mathrm{GL}_2\)-type: yes

This is a model for the modular curve $X_1(18)$. The integer $18$ is the largest $N \in \mathbb{N}$ such that $X_1(N)$ has genus $2$.

Minimal equation

$y^2 + (x^3 + x + 1)y = x^5 + 2x^4 + 2x^3 + x^2$

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(324\)	\(=\)	\( 2^{2} \cdot 3^{4} \)
Discriminant:	\( \Delta \)	\(=\)	\(-648\)	\(=\)	\( - 2^{3} \cdot 3^{4} \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(60\)	\(=\)	\( 2^{2} \cdot 3 \cdot 5 \)
\( I_4 \)	\(=\)	\(945\)	\(=\)	\( 3^{3} \cdot 5 \cdot 7 \)
\( I_6 \)	\(=\)	\(2295\)	\(=\)	\( 3^{3} \cdot 5 \cdot 17 \)
\( I_{10} \)	\(=\)	\(82944\)	\(=\)	\( 2^{10} \cdot 3^{4} \)

Automorphism group

\(\mathrm{Aut}(X)\)	\(\simeq\)	$C_6$
\(\mathrm{Aut}(X_{\overline{\Q}})\)	\(\simeq\)	$D_6$

Rational points

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

Number of rational Weierstrass points: \(0\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

Group structure: \(\Z/{21}\Z\)

Generator	$D_0$						Height	Order
\((-1 : 1 : 1) + (0 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\)	\(x (x + z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-2xz^2 - z^3\)	\(0\)	\(21\)

2-torsion field: 6.0.41472.1

BSD invariants

Hasse-Weil conjecture:	verified
Analytic rank:	\(0\)
Mordell-Weil rank:	\(0\)
2-Selmer rank:	\(0\)
Regulator:	\( 1 \)
Real period:	\( 25.52176 \)
Tamagawa product:	\( 3 \)
Torsion order:	\( 21 \)
Leading coefficient:	\( 0.173617 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

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

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?
\(2\)	2.40.3	no
\(3\)	3.1920.3	yes

Sato-Tate group

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

Decomposition of the Jacobian

Splits over the number field \(\Q (b) \simeq \) \(\Q(\zeta_{9})^+\) with defining polynomial:
  \(x^{3} - 3 x - 1\)

Decomposes up to isogeny as the square of the elliptic curve isogeny class:
  Elliptic curve isogeny class 3.3.81.1-8.1-a

Endomorphisms of the Jacobian

Of \(\GL_2\)-type over \(\Q\)

Endomorphism ring over \(\Q\):

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

Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) \(\Q(\zeta_{9})^+\) with defining polynomial \(x^{3} - 3 x - 1\)

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