Genus 2 curve 2169.a.175689.1

• Label: 2169.a.175689.1
• Conductor: $2169$
• Discriminant: $175689$
• Mordell-Weil group: \(\Z/{2}\Z \oplus \Z/{2}\Z \oplus \Z/{2}\Z\)
• 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

$y^2 + (x^2 + x)y = x^5 - 9x^4 + 22x^3 - 14x^2 - x$

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(2169\)	\(=\)	\( 3^{2} \cdot 241 \)
Discriminant:	\( \Delta \)	\(=\)	\(175689\)	\(=\)	\( 3^{6} \cdot 241 \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(2860\)	\(=\)	\( 2^{2} \cdot 5 \cdot 11 \cdot 13 \)
\( I_4 \)	\(=\)	\(62145\)	\(=\)	\( 3^{2} \cdot 5 \cdot 1381 \)
\( I_6 \)	\(=\)	\(64270095\)	\(=\)	\( 3 \cdot 5 \cdot 349 \cdot 12277 \)
\( I_{10} \)	\(=\)	\(92544\)	\(=\)	\( 2^{7} \cdot 3 \cdot 241 \)

Automorphism group

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

Rational points

All points: \((1 : 0 : 0),\, (0 : 0 : 1),\, (1 : -1 : 1),\, (4 : -10 : 1)\)

Number of rational Weierstrass points: \(4\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

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

Generator	$D_0$						Height	Order
\((1 : -1 : 1) - (1 : 0 : 0)\)	\(x - z\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-z^3\)	\(0\)	\(2\)
\((0 : 0 : 1) - (1 : 0 : 0)\)	\(x\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(0\)	\(0\)	\(2\)
\((0 : 0 : 1) + (4 : -10 : 1) - 2 \cdot(1 : 0 : 0)\)	\(x (x - 4z)\)	\(=\)	\(0,\)	\(2y\)	\(=\)	\(-5xz^2\)	\(0\)	\(2\)

2-torsion field: \(\Q(\sqrt{241}) \)

BSD invariants

Hasse-Weil conjecture:	unverified
Analytic rank:	\(0\)
Mordell-Weil rank:	\(0\)
2-Selmer rank:	\(3\)
Regulator:	\( 1 \)
Real period:	\( 10.18353 \)
Tamagawa product:	\( 4 \)
Torsion order:	\( 8 \)
Leading coefficient:	\( 0.636470 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(3\)	\(2\)	\(6\)	\(4\)	\(1 + 3 T^{2}\)
\(241\)	\(1\)	\(1\)	\(1\)	\(( 1 - T )( 1 - 18 T + 241 T^{2} )\)

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?
\(2\)	2.360.2	yes
\(3\)	3.90.2	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 \(\Q (b) \simeq \) \(\Q(\sqrt{-3}) \) with defining polynomial:
  \(x^{2} - x + 1\)

Decomposes up to isogeny as the product of the non-isogenous elliptic curve isogeny classes:
  Elliptic curve isogeny class 2.0.3.1-241.1-a
  Elliptic curve isogeny class 2.0.3.1-241.2-a

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 \(K = \Q (a) \simeq \) \(\Q(\sqrt{-3}) \) with defining polynomial \(x^{2} - x + 1\)

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

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

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