Genus 2 curve 28672.d.458752.1

• Label: 28672.d.458752.1
• Conductor: $28672$
• Discriminant: $458752$
• Mordell-Weil group: \(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
• Sato-Tate group: $N(\mathrm{SU}(2)\times\mathrm{SU}(2))$
• \(\End(J_{\overline{\Q}}) \otimes \R\): \(\R \times \R\)
• \(\End(J_{\overline{\Q}}) \otimes \Q\): \(\Q \times \Q\)
• \(\End(J) \otimes \Q\): \(\Q\)
• \(\overline{\Q}\)-simple: no
• \(\mathrm{GL}_2\)-type: no

Minimal equation

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

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(28672\)	\(=\)	\( 2^{12} \cdot 7 \)
Discriminant:	\( \Delta \)	\(=\)	\(458752\)	\(=\)	\( 2^{16} \cdot 7 \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(44\)	\(=\)	\( 2^{2} \cdot 11 \)
\( I_4 \)	\(=\)	\(76\)	\(=\)	\( 2^{2} \cdot 19 \)
\( I_6 \)	\(=\)	\(1832\)	\(=\)	\( 2^{3} \cdot 229 \)
\( I_{10} \)	\(=\)	\(56\)	\(=\)	\( 2^{3} \cdot 7 \)

Automorphism group

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

Rational points

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

Number of rational Weierstrass points: \(2\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

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

Generator	$D_0$						Height	Order
\((2 : -4 : 1) - (1 : 0 : 0)\)	\(x - 2z\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-4z^3\)	\(0.768696\)	\(\infty\)
\(D_0 - 2 \cdot(1 : 0 : 0)\)	\(x^2 - xz + 2z^2\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(0\)	\(0\)	\(2\)
\((0 : 0 : 1) - (1 : 0 : 0)\)	\(x\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(0\)	\(0\)	\(2\)

2-torsion field: \(\Q(i, \sqrt{7})\)

BSD invariants

Hasse-Weil conjecture:	unverified
Analytic rank:	\(1\)
Mordell-Weil rank:	\(1\)
2-Selmer rank:	\(3\)
Regulator:	\( 0.768696 \)
Real period:	\( 8.090658 \)
Tamagawa product:	\( 4 \)
Torsion order:	\( 4 \)
Leading coefficient:	\( 1.554814 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(2\)	\(12\)	\(16\)	\(4\)	\(1\)
\(7\)	\(1\)	\(1\)	\(1\)	\(( 1 + T )( 1 + 4 T + 7 T^{2} )\)

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
\(3\)	3.45.1	no

Sato-Tate group

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

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 2.2.8.1-448.2-b
  Elliptic curve isogeny class 2.2.8.1-448.1-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 \(K = \Q (a) \simeq \) \(\Q(\sqrt{2}) \) with defining polynomial \(x^{2} - 2\)

Of \(\GL_2\)-type over \(\overline{\Q}\)

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

\(\End (J_{\overline{\Q}})\)	\(\simeq\)	an order of index \(2\) in \(\Z \times \Z\)
\(\End (J_{\overline{\Q}}) \otimes \Q \)	\(\simeq\)	\(\Q\) \(\times\) \(\Q\)
\(\End (J_{\overline{\Q}}) \otimes \R\)	\(\simeq\)	\(\R \times \R\)