Properties

Label 10080.a.60480.1
Conductor $10080$
Discriminant $60480$
Mordell-Weil group \(\Z/{2}\Z \times \Z/{2}\Z\)
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

Related objects

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Show commands: SageMath / Magma

Minimal equation

Minimal equation

Simplified equation

$y^2 + xy = -15x^6 + 58x^4 - 60x^2 + 7$ (homogenize, simplify)
$y^2 + xz^2y = -15x^6 + 58x^4z^2 - 60x^2z^4 + 7z^6$ (dehomogenize, simplify)
$y^2 = -60x^6 + 232x^4 - 239x^2 + 28$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([7, 0, -60, 0, 58, 0, -15]), R([0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![7, 0, -60, 0, 58, 0, -15], R![0, 1]);
 
sage: X = HyperellipticCurve(R([28, 0, -239, 0, 232, 0, -60]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(10080\) \(=\) \( 2^{5} \cdot 3^{2} \cdot 5 \cdot 7 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(60480\) \(=\) \( 2^{6} \cdot 3^{3} \cdot 5 \cdot 7 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(161296\) \(=\)  \( 2^{4} \cdot 17 \cdot 593 \)
\( I_4 \)  \(=\) \(406586887\) \(=\)  \( 7 \cdot 43 \cdot 67 \cdot 20161 \)
\( I_6 \)  \(=\) \(19127473723714\) \(=\)  \( 2 \cdot 347 \cdot 27561201331 \)
\( I_{10} \)  \(=\) \(7560\) \(=\)  \( 2^{3} \cdot 3^{3} \cdot 5 \cdot 7 \)
\( J_2 \)  \(=\) \(161296\) \(=\)  \( 2^{4} \cdot 17 \cdot 593 \)
\( J_4 \)  \(=\) \(812958726\) \(=\)  \( 2 \cdot 3 \cdot 2399 \cdot 56479 \)
\( J_6 \)  \(=\) \(4856153621760\) \(=\)  \( 2^{8} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 89 \cdot 151 \cdot 4481 \)
\( J_8 \)  \(=\) \(30594066098964471\) \(=\)  \( 3^{2} \cdot 191 \cdot 311 \cdot 57226994119 \)
\( J_{10} \)  \(=\) \(60480\) \(=\)  \( 2^{6} \cdot 3^{3} \cdot 5 \cdot 7 \)
\( g_1 \)  \(=\) \(1705838896690345318825984/945\)
\( g_2 \)  \(=\) \(17767980154611986862208/315\)
\( g_3 \)  \(=\) \(6266846885932235776/3\)

sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]
 
magma: IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);
 

Automorphism group

\(\mathrm{Aut}(X)\)\(\simeq\) $C_2^2$
magma: AutomorphismGroup(C); IdentifyGroup($1);
 
\(\mathrm{Aut}(X_{\overline{\Q}})\)\(\simeq\) $C_2^2$
magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
 

Rational points

This curve has no rational points.
This curve has no rational points.
This curve has no rational points.

magma: []; // minimal model
 
magma: []; // simplified model
 

Number of rational Weierstrass points: \(0\)

magma: #Roots(HyperellipticPolynomials(SimplifiedModel(C)));
 

This curve is locally solvable everywhere.

magma: f,h:=HyperellipticPolynomials(C); g:=4*f+h^2; HasPointsEverywhereLocally(g,2) and (#Roots(ChangeRing(g,RealField())) gt 0 or LeadingCoefficient(g) gt 0);
 

Mordell-Weil group of the Jacobian

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

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - D_\infty\) \(15x^2 - 28z^2\) \(=\) \(0,\) \(2y\) \(=\) \(-xz^2\) \(0\) \(2\)
\(D_0 - D_\infty\) \(2x^2 - 2xz - z^2\) \(=\) \(0,\) \(2y\) \(=\) \(-xz^2\) \(0\) \(2\)
Generator $D_0$ Height Order
\(D_0 - D_\infty\) \(15x^2 - 28z^2\) \(=\) \(0,\) \(2y\) \(=\) \(-xz^2\) \(0\) \(2\)
\(D_0 - D_\infty\) \(2x^2 - 2xz - z^2\) \(=\) \(0,\) \(2y\) \(=\) \(-xz^2\) \(0\) \(2\)
Generator $D_0$ Height Order
\(D_0 - D_\infty\) \(15x^2 - 28z^2\) \(=\) \(0,\) \(2y\) \(=\) \(-xz^2\) \(0\) \(2\)
\(D_0 - D_\infty\) \(2x^2 - 2xz - z^2\) \(=\) \(0,\) \(2y\) \(=\) \(-xz^2\) \(0\) \(2\)

2-torsion field: \(\Q(\sqrt{3}, \sqrt{35})\)

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(4\)
Regulator: \( 1 \)
Real period: \( 0.556034 \)
Tamagawa product: \( 6 \)
Torsion order:\( 4 \)
Leading coefficient: \( 0.834051 \)
Analytic order of Ш: \( 4 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(5\) \(6\) \(2\) \(1 + T + 2 T^{2}\)
\(3\) \(2\) \(3\) \(3\) \(( 1 - T )^{2}\)
\(5\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 2 T + 5 T^{2} )\)
\(7\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 - 4 T + 7 T^{2} )\)

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 480.f
  Elliptic curve isogeny class 21.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 \(\Q\).