Genus 2 curve 864.a.221184.1

• Label: 864.a.221184.1
• Conductor: $864$
• Discriminant: $-221184$
• Mordell-Weil group: \(\Z/{12}\Z\)
• Sato-Tate group: $N(\mathrm{U}(1)\times\mathrm{SU}(2))$
• \(\End(J_{\overline{\Q}}) \otimes \R\): \(\C \times \R\)
• \(\End(J_{\overline{\Q}}) \otimes \Q\): \(\mathsf{CM} \times \Q\)
• \(\End(J) \otimes \Q\): \(\Q \times \Q\)
• \(\overline{\Q}\)-simple: no
• \(\mathrm{GL}_2\)-type: yes

Minimal equation

$y^2 + x^3y = x^5 - 4x^4 - 6x^3 + 33x^2 - 36x + 12$

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(864\)	\(=\)	\( 2^{5} \cdot 3^{3} \)
Discriminant:	\( \Delta \)	\(=\)	\(-221184\)	\(=\)	\( - 2^{13} \cdot 3^{3} \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(168\)	\(=\)	\( 2^{3} \cdot 3 \cdot 7 \)
\( I_4 \)	\(=\)	\(34560\)	\(=\)	\( 2^{8} \cdot 3^{3} \cdot 5 \)
\( I_6 \)	\(=\)	\(-211428\)	\(=\)	\( - 2^{2} \cdot 3^{2} \cdot 7 \cdot 839 \)
\( I_{10} \)	\(=\)	\(-864\)	\(=\)	\( - 2^{5} \cdot 3^{3} \)

Automorphism group

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

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: \(\Z/{12}\Z\)

Generator	$D_0$						Height	Order
\(2 \cdot(1 : 0 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\)	\((x - z)^2\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(xz^2 - z^3\)	\(0\)	\(12\)

2-torsion field: 8.0.47775744.4

BSD invariants

Hasse-Weil conjecture:	verified
Analytic rank:	\(0\)
Mordell-Weil rank:	\(0\)
2-Selmer rank:	\(1\)
Regulator:	\( 1 \)
Real period:	\( 18.14296 \)
Tamagawa product:	\( 3 \)
Torsion order:	\( 12 \)
Leading coefficient:	\( 0.377978 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(2\)	\(5\)	\(13\)	\(3\)	\(1\)
\(3\)	\(3\)	\(3\)	\(1\)	\(1 + T\)

Sato-Tate group

\(\mathrm{ST}\)	\(\simeq\)	$N(\mathrm{U}(1)\times\mathrm{SU}(2))$
\(\mathrm{ST}^0\)	\(\simeq\)	\(\mathrm{U}(1)\times\mathrm{SU}(2)\)

Decomposition of the Jacobian

Splits over \(\Q\)

Decomposes up to isogeny as the product of the non-isogenous elliptic curves:
  Elliptic curve 36.a2
  Elliptic curve 24.a2

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 \(K = \Q (a) \simeq \) \(\Q(\sqrt{-3}) \) with defining polynomial \(x^{2} - x + 1\)

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

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

\(\End (J_{\overline{\Q}})\)	\(\simeq\)	an order of index \(8\) in \(\Z \times \Z [\frac{1 + \sqrt{-3}}{2}]\)
\(\End (J_{\overline{\Q}}) \otimes \Q \)	\(\simeq\)	\(\Q\) \(\times\) \(\Q(\sqrt{-3}) \)
\(\End (J_{\overline{\Q}}) \otimes \R\)	\(\simeq\)	\(\R \times \C\)