Properties

Label 720.a.6480.1
Conductor $720$
Discriminant $-6480$
Mordell-Weil group \(\Z/{2}\Z \times \Z/{4}\Z\)
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 for: SageMath / Magma

Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^3 + x)y = 2x^4 + 7x^2 + 5$ (homogenize, simplify)
$y^2 + (x^3 + xz^2)y = 2x^4z^2 + 7x^2z^4 + 5z^6$ (dehomogenize, simplify)
$y^2 = x^6 + 10x^4 + 29x^2 + 20$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([5, 0, 7, 0, 2]), R([0, 1, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![5, 0, 7, 0, 2], R![0, 1, 0, 1]);
 
sage: X = HyperellipticCurve(R([20, 0, 29, 0, 10, 0, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(720\) \(=\) \( 2^{4} \cdot 3^{2} \cdot 5 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-6480\) \(=\) \( - 2^{4} \cdot 3^{4} \cdot 5 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(2360\) \(=\)  \( 2^{3} \cdot 5 \cdot 59 \)
\( I_4 \)  \(=\) \(11992\) \(=\)  \( 2^{3} \cdot 1499 \)
\( I_6 \)  \(=\) \(9047820\) \(=\)  \( 2^{2} \cdot 3 \cdot 5 \cdot 150797 \)
\( I_{10} \)  \(=\) \(25920\) \(=\)  \( 2^{6} \cdot 3^{4} \cdot 5 \)
\( J_2 \)  \(=\) \(1180\) \(=\)  \( 2^{2} \cdot 5 \cdot 59 \)
\( J_4 \)  \(=\) \(56018\) \(=\)  \( 2 \cdot 37 \cdot 757 \)
\( J_6 \)  \(=\) \(3453120\) \(=\)  \( 2^{6} \cdot 3^{2} \cdot 5 \cdot 11 \cdot 109 \)
\( J_8 \)  \(=\) \(234166319\) \(=\)  \( 3299 \cdot 70981 \)
\( J_{10} \)  \(=\) \(6480\) \(=\)  \( 2^{4} \cdot 3^{4} \cdot 5 \)
\( g_1 \)  \(=\) \(28596971960000/81\)
\( g_2 \)  \(=\) \(1150492082200/81\)
\( g_3 \)  \(=\) \(6677950400/9\)

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

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

magma: [C![1,-1,0],C![1,0,0]];
 

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: \(\Z/{2}\Z \times \Z/{4}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

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

2-torsion field: \(\Q(i, \sqrt{5})\)

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(2\)
Regulator: \( 1 \)
Real period: \( 9.444268 \)
Tamagawa product: \( 2 \)
Torsion order:\( 8 \)
Leading coefficient: \( 0.295133 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

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

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 curves:
  Elliptic curve 48.a4
  Elliptic curve 15.a6

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 \(\Q\).