Genus 2 curve 5329.b.5329.1

• Label: 5329.b.5329.1
• Conductor: $5329$
• Discriminant: $5329$
• Mordell-Weil group: \(\Z \oplus \Z\)
• Sato-Tate group: $\mathrm{SU}(2)\times\mathrm{SU}(2)$
• \(\End(J_{\overline{\Q}}) \otimes \R\): \(\R \times \R\)
• \(\End(J_{\overline{\Q}}) \otimes \Q\): \(\mathsf{RM}\)
• \(\End(J) \otimes \Q\): \(\mathsf{RM}\)
• \(\overline{\Q}\)-simple: yes
• \(\mathrm{GL}_2\)-type: yes

This is a model for the quotient of the modular curve $X_0(73)$ by its Fricke involution $w_{73}$; this quotient is also denoted $X_0^+(73)$.

Minimal equation

$y^2 + (x^3 + x^2 + 1)y = x^3 - x$

(, )

Invariants

Conductor:	\( N \)	\(=\)	\(5329\)	\(=\)	\( 73^{2} \)
Discriminant:	\( \Delta \)	\(=\)	\(5329\)	\(=\)	\( 73^{2} \)

Igusa-Clebsch invariants

\( I_2 \)	\(=\)	\(188\)	\(=\)	\( 2^{2} \cdot 47 \)
\( I_4 \)	\(=\)	\(3721\)	\(=\)	\( 61^{2} \)
\( I_6 \)	\(=\)	\(413963\)	\(=\)	\( 11 \cdot 37633 \)
\( I_{10} \)	\(=\)	\(-682112\)	\(=\)	\( - 2^{7} \cdot 73^{2} \)

Automorphism group

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

Rational points

All points
\((1 : 0 : 0)\)	\((1 : -1 : 0)\)	\((0 : 0 : 1)\)	\((-1 : 0 : 1)\)	\((0 : -1 : 1)\)	\((1 : 0 : 1)\)
\((-1 : -1 : 1)\)	\((1 : -3 : 1)\)	\((1 : -3 : 2)\)	\((1 : -8 : 2)\)

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
\((0 : 0 : 1) - (1 : 0 : 0)\)	\(z x\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-x^3\)	\(0.136589\)	\(\infty\)
\((0 : -1 : 1) - (1 : 0 : 0)\)	\(z x\)	\(=\)	\(0,\)	\(y\)	\(=\)	\(-x^3 - z^3\)	\(0.092601\)	\(\infty\)

2-torsion field: 5.1.341056.1

BSD invariants

Hasse-Weil conjecture:	verified
Analytic rank:	\(2\)
Mordell-Weil rank:	\(2\)
2-Selmer rank:	\(2\)
Regulator:	\( 0.010713 \)
Real period:	\( 24.09331 \)
Tamagawa product:	\( 1 \)
Torsion order:	\( 1 \)
Leading coefficient:	\( 0.258121 \)
Analytic order of Ш:	\( 1 \)   (rounded)
Order of Ш:	square

Local invariants

Prime	ord(\(N\))	ord(\(\Delta\))	Tamagawa	L-factor	Cluster picture
\(73\)	\(2\)	\(2\)	\(1\)	\(( 1 + T )^{2}\)

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?
\(2\)	2.12.2	no
\(3\)	3.2160.18	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

Simple over \(\overline{\Q}\)

Endomorphisms of the Jacobian

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

Endomorphism ring over \(\Q\):

\(\End (J_{})\)	\(\simeq\)	\(\Z [\frac{1 + \sqrt{5}}{2}]\)
\(\End (J_{}) \otimes \Q \)	\(\simeq\)	\(\Q(\sqrt{5}) \)
\(\End (J_{}) \otimes \R\)	\(\simeq\)	\(\R \times \R\)

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 Balakrishnan--Best--Bianchi--Lawrence--Müller--Triantafillou--Vonk using quadratic Chabauty and the Mordell--Weil sieve.