Genus 2 curve 5547.b.16641.1

• Label: 5547.b.16641.1
• Conductor: $5547$
• Discriminant: $16641$
• Mordell-Weil group: \(\Z \oplus \Z\)
• 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

This is a model for the quotient of the modular curve $X_0(129)$ by the action of the group of order 4 generated by the Atkin-Lehner involutions $w_3$ and $w_{43}$. This quotient is also denoted $X_0^*(129)$.

Minimal equation

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

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(5547\)	\(=\)	\( 3 \cdot 43^{2} \)
Discriminant:	\( \Delta \)	\(=\)	\(16641\)	\(=\)	\( 3^{2} \cdot 43^{2} \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(520\)	\(=\)	\( 2^{3} \cdot 5 \cdot 13 \)
\( I_4 \)	\(=\)	\(6292\)	\(=\)	\( 2^{2} \cdot 11^{2} \cdot 13 \)
\( I_6 \)	\(=\)	\(896816\)	\(=\)	\( 2^{4} \cdot 23 \cdot 2437 \)
\( I_{10} \)	\(=\)	\(66564\)	\(=\)	\( 2^{2} \cdot 3^{2} \cdot 43^{2} \)

Automorphism group

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

Rational points

All points
\((1 : -1 : 0)\)	\((1 : 1 : 0)\)	\((0 : 0 : 1)\)	\((0 : -1 : 1)\)	\((1 : 0 : 1)\)	\((-1 : 1 : 1)\)
\((1 : -1 : 1)\)	\((-1 : -2 : 1)\)	\((2 : 1 : 1)\)	\((2 : -2 : 1)\)	\((-5 : 20 : 7)\)	\((12 : 20 : 7)\)
\((-5 : -363 : 7)\)	\((12 : -363 : 7)\)

Number of rational Weierstrass points: \(0\)

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian

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

Generator	$D_0$						Height	Order
\((0 : 0 : 1) + (2 : -2 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\)	\(x (x - 2z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-xz^2\)	\(0.081387\)	\(\infty\)
\((-1 : -2 : 1) + (1 : 0 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\)	\((x - z) (x + z)\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(xz^2 - z^3\)	\(0.081387\)	\(\infty\)

2-torsion field: 4.2.688.1

BSD invariants

Hasse-Weil conjecture:	verified
Analytic rank:	\(2\)
Mordell-Weil rank:	\(2\)
2-Selmer rank:	\(2\)
Regulator:	\( 0.006279 \)
Real period:	\( 24.46037 \)
Tamagawa product:	\( 2 \)
Torsion order:	\( 1 \)
Leading coefficient:	\( 0.307177 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	Root number	L-factor	Cluster picture	Tame reduction?
\(3\)	\(1\)	\(2\)	\(2\)	\(1\)	\(( 1 + T )( 1 + 2 T + 3 T^{2} )\)		yes
\(43\)	\(2\)	\(2\)	\(1\)	\(1\)	\(( 1 + T )^{2}\)		yes

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
\(3\)	3.90.1	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 129.a
  Elliptic curve isogeny class 43.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 \(\Q\).

Additional information

The set of rational points on this genus 2 curve with rank 2 Jacobian was computed by Bars--González--Xarles using 2-cover descent and elliptic Chabauty and Edixhoven--Lido using geometric quadratic Chabauty.