Genus 2 curve 1680.a.16800.1

• Label: 1680.a.16800.1
• Conductor: $1680$
• Discriminant: $-16800$
• Mordell-Weil group: \(\Z/{2}\Z \oplus \Z/{4}\Z\)
• Sato-Tate group: $\mathrm{SU}(2)\times\mathrm{SU}(2)$
• \(\End(J_{\overline{\Q}}) \otimes \R\): \(\R \times \R\)
• \(\End(J_{\overline{\Q}}) \otimes \Q\): \(\Q \times \Q\)
• \(\End(J) \otimes \Q\): \(\Q \times \Q\)
• \(\overline{\Q}\)-simple: no
• \(\mathrm{GL}_2\)-type: yes

Minimal equation

$y^2 + (x^3 + x)y = -x^6 - 18x^4 - 136x^2 - 350$

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(1680\)	\(=\)	\( 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Discriminant:	\( \Delta \)	\(=\)	\(-16800\)	\(=\)	\( - 2^{5} \cdot 3 \cdot 5^{2} \cdot 7 \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(404040\)	\(=\)	\( 2^{3} \cdot 3 \cdot 5 \cdot 7 \cdot 13 \cdot 37 \)
\( I_4 \)	\(=\)	\(44088\)	\(=\)	\( 2^{3} \cdot 3 \cdot 11 \cdot 167 \)
\( I_6 \)	\(=\)	\(5935895700\)	\(=\)	\( 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 1319 \cdot 2143 \)
\( I_{10} \)	\(=\)	\(67200\)	\(=\)	\( 2^{7} \cdot 3 \cdot 5^{2} \cdot 7 \)

Automorphism group

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

Rational points

This curve has no rational points.

Number of rational Weierstrass points: \(0\)

This curve is locally solvable except over $\R$.

Mordell-Weil group of the Jacobian

Group structure: \(\Z/{2}\Z \oplus \Z/{4}\Z\)

Generator	$D_0$						Height	Order
\(D_0 - D_\infty\)	\(x^2 + 8z^2\)	\(=\)	\(0,\)	\(2y\)	\(=\)	\(7xz^2\)	\(0\)	\(2\)
\(D_0 - D_\infty\)	\(x^2 + 9z^2\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(4xz^2 + z^3\)	\(0\)	\(4\)

2-torsion field: 8.0.796594176.1

BSD invariants

Hasse-Weil conjecture:	verified
Analytic rank:	\(0\)
Mordell-Weil rank:	\(0\)
2-Selmer rank:	\(3\)
Regulator:	\( 1 \)
Real period:	\( 5.090689 \)
Tamagawa product:	\( 4 \)
Torsion order:	\( 8 \)
Leading coefficient:	\( 0.636336 \)
Analytic order of Ш:	\( 2 \)   (rounded)
Order of Ш:	twice a square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(2\)	\(4\)	\(5\)	\(2\)	\(1 - T\)
\(3\)	\(1\)	\(1\)	\(1\)	\(( 1 + T )( 1 + 3 T^{2} )\)
\(5\)	\(1\)	\(2\)	\(2\)	\(( 1 + T )( 1 + 2 T + 5 T^{2} )\)
\(7\)	\(1\)	\(1\)	\(1\)	\(( 1 + T )( 1 + 7 T^{2} )\)

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?
\(2\)	2.90.6	yes
\(3\)	3.90.1	no

Sato-Tate group

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

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 24.a
  Elliptic curve isogeny class 70.a

Endomorphisms of the Jacobian

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

Endomorphism ring over \(\Q\):

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

All \(\overline{\Q}\)-endomorphisms of the Jacobian are defined over \(\Q\).