Genus 2 curve 3072.a.196608.2

• Label: 3072.a.196608.2
• Conductor: $3072$
• Discriminant: $196608$
• Mordell-Weil group: \(\Z/{2}\Z \oplus \Z/{2}\Z\)
• 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

$y^2 = 6x^6 - 13x^4 + 9x^2 - 2$

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(3072\)	\(=\)	\( 2^{10} \cdot 3 \)
Discriminant:	\( \Delta \)	\(=\)	\(196608\)	\(=\)	\( 2^{16} \cdot 3 \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(2376\)	\(=\)	\( 2^{3} \cdot 3^{3} \cdot 11 \)
\( I_4 \)	\(=\)	\(321\)	\(=\)	\( 3 \cdot 107 \)
\( I_6 \)	\(=\)	\(254043\)	\(=\)	\( 3^{3} \cdot 97^{2} \)
\( I_{10} \)	\(=\)	\(24\)	\(=\)	\( 2^{3} \cdot 3 \)

Automorphism group

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

Rational points

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

Number of rational Weierstrass points: \(2\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

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

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

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

BSD invariants

Hasse-Weil conjecture:	verified
Analytic rank:	\(0\)
Mordell-Weil rank:	\(0\)
2-Selmer rank:	\(2\)
Regulator:	\( 1 \)
Real period:	\( 10.49953 \)
Tamagawa product:	\( 1 \)
Torsion order:	\( 4 \)
Leading coefficient:	\( 0.656220 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(2\)	\(10\)	\(16\)	\(1\)	\(1\)
\(3\)	\(1\)	\(1\)	\(1\)	\(( 1 + T )( 1 + 3 T^{2} )\)

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?
\(2\)	2.180.3	yes
\(3\)	3.270.2	no

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 curve isogeny classes:
  Elliptic curve isogeny class 32.a
  Elliptic curve isogeny class 96.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 \(K = \Q (a) \simeq \) \(\Q(\sqrt{-1}) \) with defining polynomial \(x^{2} + 1\)

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

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

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