Genus 2 curve 994009.a.994009.1

• Label: 994009.a.994009.1
• Conductor: $994009$
• Discriminant: $994009$
• Mordell-Weil group: \(\Z \oplus \Z \oplus \Z \oplus \Z\)
• 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

$y^2 + x^3y = -4x^4 - 7x^3 - x^2 + 3x + 1$

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(994009\)	\(=\)	\( 997^{2} \)
Discriminant:	\( \Delta \)	\(=\)	\(994009\)	\(=\)	\( 997^{2} \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(680\)	\(=\)	\( 2^{3} \cdot 5 \cdot 17 \)
\( I_4 \)	\(=\)	\(14932\)	\(=\)	\( 2^{2} \cdot 3733 \)
\( I_6 \)	\(=\)	\(2967104\)	\(=\)	\( 2^{6} \cdot 7 \cdot 37 \cdot 179 \)
\( I_{10} \)	\(=\)	\(3976036\)	\(=\)	\( 2^{2} \cdot 997^{2} \)

Automorphism group

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

Rational points

Known points
\((1 : 0 : 0)\)	\((1 : -1 : 0)\)	\((-1 : 0 : 1)\)	\((0 : -1 : 1)\)	\((0 : 1 : 1)\)	\((-1 : 1 : 1)\)
\((1 : 8 : 2)\)	\((-3 : 8 : 1)\)	\((-1 : -8 : 3)\)	\((-3 : 8 : 2)\)	\((1 : -9 : 2)\)	\((-1 : 9 : 3)\)
\((-3 : 19 : 1)\)	\((-3 : 19 : 2)\)	\((-5 : 53 : 2)\)	\((-5 : 53 : 3)\)	\((-5 : 72 : 2)\)	\((-5 : 72 : 3)\)

Number of rational Weierstrass points: \(0\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

Group structure: \(\Z \oplus \Z \oplus \Z \oplus \Z\)

Generator	$D_0$						Height	Order
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\)	\(x^2 + xz - z^2\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-xz^2 + z^3\)	\(0.949694\)	\(\infty\)
\((-1 : 0 : 1) - (1 : 0 : 0)\)	\(z (x + z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-x^3 - z^3\)	\(0.399356\)	\(\infty\)
\((-1 : 1 : 1) + (0 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\)	\(x (x + z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-2xz^2 - z^3\)	\(0.801047\)	\(\infty\)
\((-1 : 0 : 1) - (1 : -1 : 0)\)	\(z (x + z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(0\)	\(0.691172\)	\(\infty\)

2-torsion field: 4.4.15952.1

BSD invariants

Hasse-Weil conjecture:	verified
Analytic rank:	\(4\)
Mordell-Weil rank:	\(4\)
2-Selmer rank:	\(4\)
Regulator:	\( 0.142586 \)
Real period:	\( 15.40967 \)
Tamagawa product:	\( 1 \)
Torsion order:	\( 1 \)
Leading coefficient:	\( 2.197213 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(997\)	\(2\)	\(2\)	\(1\)	\(( 1 - T )^{2}\)

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?
\(2\)	2.30.2	no
\(3\)	3.270.3	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 997.c
  Elliptic curve isogeny class 997.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 \(\Q\).