Properties

Label 960.a.245760.1
Conductor $960$
Discriminant $245760$
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

Learn more about

Show commands for: SageMath / Magma

Minimal equation

Minimal equation

Simplified equation

$y^2 = 2x^5 + x^4 + 4x^3 + x^2 + 2x$ (homogenize, simplify)
$y^2 = 2x^5z + x^4z^2 + 4x^3z^3 + x^2z^4 + 2xz^5$ (dehomogenize, simplify)
$y^2 = 2x^5 + x^4 + 4x^3 + x^2 + 2x$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 2, 1, 4, 1, 2]), R([]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 2, 1, 4, 1, 2], R![]);
 
sage: X = HyperellipticCurve(R([0, 2, 1, 4, 1, 2]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(960\) \(=\) \( 2^{6} \cdot 3 \cdot 5 \)
magma: Conductor(LSeries(C: ExcFactors:=[*<2,Valuation(960,2),R![1]>*])); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(245760\) \(=\) \( 2^{14} \cdot 3 \cdot 5 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(120\) \(=\)  \( 2^{3} \cdot 3 \cdot 5 \)
\( I_4 \)  \(=\) \(213\) \(=\)  \( 3 \cdot 71 \)
\( I_6 \)  \(=\) \(10095\) \(=\)  \( 3 \cdot 5 \cdot 673 \)
\( I_{10} \)  \(=\) \(30\) \(=\)  \( 2 \cdot 3 \cdot 5 \)
\( J_2 \)  \(=\) \(480\) \(=\)  \( 2^{5} \cdot 3 \cdot 5 \)
\( J_4 \)  \(=\) \(7328\) \(=\)  \( 2^{5} \cdot 229 \)
\( J_6 \)  \(=\) \(-15360\) \(=\)  \( - 2^{10} \cdot 3 \cdot 5 \)
\( J_8 \)  \(=\) \(-15268096\) \(=\)  \( - 2^{8} \cdot 19 \cdot 43 \cdot 73 \)
\( J_{10} \)  \(=\) \(245760\) \(=\)  \( 2^{14} \cdot 3 \cdot 5 \)
\( g_1 \)  \(=\) \(103680000\)
\( g_2 \)  \(=\) \(3297600\)
\( g_3 \)  \(=\) \(-14400\)

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

magma: [C![0,0,1],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
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 + z^2\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(2\)
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 + xz + z^2\) \(=\) \(0,\) \(y\) \(=\) \(z^3\) \(0\) \(4\)

2-torsion field: \(\Q(i, \sqrt{15})\)

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(2\)
Regulator: \( 1 \)
Real period: \( 6.402317 \)
Tamagawa product: \( 4 \)
Torsion order:\( 8 \)
Leading coefficient: \( 0.400144 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(6\) \(14\) \(4\) \(1\)
\(3\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 3 T^{2} )\)
\(5\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 2 T + 5 T^{2} )\)

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 40.a2
  Elliptic curve 24.a4

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 \(\Q\).