Properties

Label 930.a.930.1
Conductor $930$
Discriminant $930$
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

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Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^2 + x)y = -x^5 - 7x^4 + 37x^2 - 45x + 15$ (homogenize, simplify)
$y^2 + (x^2z + xz^2)y = -x^5z - 7x^4z^2 + 37x^2z^4 - 45xz^5 + 15z^6$ (dehomogenize, simplify)
$y^2 = -4x^5 - 27x^4 + 2x^3 + 149x^2 - 180x + 60$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([15, -45, 37, 0, -7, -1]), R([0, 1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![15, -45, 37, 0, -7, -1], R![0, 1, 1]);
 
sage: X = HyperellipticCurve(R([60, -180, 149, 2, -27, -4]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(930\) \(=\) \( 2 \cdot 3 \cdot 5 \cdot 31 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(930\) \(=\) \( 2 \cdot 3 \cdot 5 \cdot 31 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(46596\) \(=\)  \( 2^{2} \cdot 3 \cdot 11 \cdot 353 \)
\( I_4 \)  \(=\) \(239073\) \(=\)  \( 3 \cdot 79691 \)
\( I_6 \)  \(=\) \(3674852529\) \(=\)  \( 3 \cdot 19 \cdot 751 \cdot 85847 \)
\( I_{10} \)  \(=\) \(119040\) \(=\)  \( 2^{8} \cdot 3 \cdot 5 \cdot 31 \)
\( J_2 \)  \(=\) \(11649\) \(=\)  \( 3 \cdot 11 \cdot 353 \)
\( J_4 \)  \(=\) \(5644172\) \(=\)  \( 2^{2} \cdot 1411043 \)
\( J_6 \)  \(=\) \(3640360380\) \(=\)  \( 2^{2} \cdot 3 \cdot 5 \cdot 31 \cdot 1249 \cdot 1567 \)
\( J_8 \)  \(=\) \(2637470125259\) \(=\)  \( 64399 \cdot 40955141 \)
\( J_{10} \)  \(=\) \(930\) \(=\)  \( 2 \cdot 3 \cdot 5 \cdot 31 \)
\( g_1 \)  \(=\) \(71502622649365111083/310\)
\( g_2 \)  \(=\) \(1487013548016809538/155\)
\( g_3 \)  \(=\) \(531176338621566\)

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

magma: [C![1,-1,1],C![1,0,0]];
 

Number of rational Weierstrass points: \(2\)

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

2-torsion field: \(\Q(\sqrt{2}, \sqrt{465})\)

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(2\)
Regulator: \( 1 \)
Real period: \( 24.84648 \)
Tamagawa product: \( 1 \)
Torsion order:\( 8 \)
Leading coefficient: \( 0.388226 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + T + 2 T^{2} )\)
\(3\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 3 T^{2} )\)
\(5\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 2 T + 5 T^{2} )\)
\(31\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 31 T^{2} )\)

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 15.a6
  Elliptic curve 62.a3

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 \(\Q\).