Genus 2 curve 21090.a.21090.1

• Label: 21090.a.21090.1
• Conductor: $21090$
• Discriminant: $-21090$
• Mordell-Weil group: \(\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
• 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

$y^2 + (x^2 + x)y = -21x^6 + 38x^5 - 86x^4 + 80x^3 - 86x^2 + 38x - 21$

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(21090\)	\(=\)	\( 2 \cdot 3 \cdot 5 \cdot 19 \cdot 37 \)
Discriminant:	\( \Delta \)	\(=\)	\(-21090\)	\(=\)	\( - 2 \cdot 3 \cdot 5 \cdot 19 \cdot 37 \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(1014780\)	\(=\)	\( 2^{2} \cdot 3 \cdot 5 \cdot 13 \cdot 1301 \)
\( I_4 \)	\(=\)	\(378177\)	\(=\)	\( 3 \cdot 37 \cdot 3407 \)
\( I_6 \)	\(=\)	\(127856778495\)	\(=\)	\( 3 \cdot 5 \cdot 10303 \cdot 827311 \)
\( I_{10} \)	\(=\)	\(2699520\)	\(=\)	\( 2^{8} \cdot 3 \cdot 5 \cdot 19 \cdot 37 \)

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

Generator	$D_0$						Height	Order
\(D_0 - D_\infty\)	\(x^2 + z^2\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(xz^2 - z^3\)	\(2.419964\)	\(\infty\)
\(D_0 - D_\infty\)	\(2x^2 - xz + 2z^2\)	\(=\)	\(0,\)	\(4y\)	\(=\)	\(-3xz^2 + 2z^3\)	\(0\)	\(2\)
\(D_0 - D_\infty\)	\(3x^2 - 2xz + 3z^2\)	\(=\)	\(0,\)	\(6y\)	\(=\)	\(-5xz^2 + 3z^3\)	\(0\)	\(2\)

2-torsion field: 8.0.50646132198812160000.1

BSD invariants

Hasse-Weil conjecture:	verified
Analytic rank:	\(1\)
Mordell-Weil rank:	\(1\)
2-Selmer rank:	\(4\)
Regulator:	\( 2.419964 \)
Real period:	\( 4.648465 \)
Tamagawa product:	\( 1 \)
Torsion order:	\( 4 \)
Leading coefficient:	\( 1.406140 \)
Analytic order of Ш:	\( 2 \)   (rounded)
Order of Ш:	twice a square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(2\)	\(1\)	\(1\)	\(1\)	\(( 1 + T )( 1 - T + 2 T^{2} )\)
\(3\)	\(1\)	\(1\)	\(1\)	\(( 1 - T )( 1 + 3 T^{2} )\)
\(5\)	\(1\)	\(1\)	\(1\)	\(( 1 + T )( 1 + 2 T + 5 T^{2} )\)
\(19\)	\(1\)	\(1\)	\(1\)	\(( 1 + T )( 1 + 4 T + 19 T^{2} )\)
\(37\)	\(1\)	\(1\)	\(1\)	\(( 1 + T )( 1 + 10 T + 37 T^{2} )\)

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 370.b
  Elliptic curve isogeny class 57.c

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 \(\Q\).