Properties

Label 864.a.442368.1
Conductor $864$
Discriminant $442368$
Mordell-Weil group \(\Z/{12}\Z\)
Sato-Tate group $N(\mathrm{U}(1)\times\mathrm{SU}(2))$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\C \times \R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathsf{CM} \times \Q\)
\(\End(J) \otimes \Q\) \(\Q \times \Q\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type yes

Related objects

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Minimal equation

Minimal equation

Simplified equation

$y^2 = x^6 - 4x^4 + 6x^2 - 3$ (homogenize, simplify)
$y^2 = x^6 - 4x^4z^2 + 6x^2z^4 - 3z^6$ (dehomogenize, simplify)
$y^2 = x^6 - 4x^4 + 6x^2 - 3$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-3, 0, 6, 0, -4, 0, 1]), R([]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-3, 0, 6, 0, -4, 0, 1], R![]);
 
sage: X = HyperellipticCurve(R([-3, 0, 6, 0, -4, 0, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(864\) \(=\) \( 2^{5} \cdot 3^{3} \)
magma: Conductor(LSeries(C: ExcFactors:=[*<2,Valuation(864,2),R![1]>*])); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(442368\) \(=\) \( 2^{14} \cdot 3^{3} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(552\) \(=\)  \( 2^{3} \cdot 3 \cdot 23 \)
\( I_4 \)  \(=\) \(45\) \(=\)  \( 3^{2} \cdot 5 \)
\( I_6 \)  \(=\) \(7083\) \(=\)  \( 3^{2} \cdot 787 \)
\( I_{10} \)  \(=\) \(54\) \(=\)  \( 2 \cdot 3^{3} \)
\( J_2 \)  \(=\) \(2208\) \(=\)  \( 2^{5} \cdot 3 \cdot 23 \)
\( J_4 \)  \(=\) \(202656\) \(=\)  \( 2^{5} \cdot 3 \cdot 2111 \)
\( J_6 \)  \(=\) \(24809472\) \(=\)  \( 2^{12} \cdot 3^{2} \cdot 673 \)
\( J_8 \)  \(=\) \(3427464960\) \(=\)  \( 2^{8} \cdot 3^{2} \cdot 5 \cdot 297523 \)
\( J_{10} \)  \(=\) \(442368\) \(=\)  \( 2^{14} \cdot 3^{3} \)
\( g_1 \)  \(=\) \(118634674176\)
\( g_2 \)  \(=\) \(4931431104\)
\( g_3 \)  \(=\) \(273421056\)

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: \((1 : -1 : 0),\, (1 : 1 : 0),\, (-1 : 0 : 1),\, (1 : 0 : 1)\)
All points: \((1 : -1 : 0),\, (1 : 1 : 0),\, (-1 : 0 : 1),\, (1 : 0 : 1)\)
All points: \((1 : -1/2 : 0),\, (1 : 1/2 : 0),\, (-1 : 0 : 1),\, (1 : 0 : 1)\)

magma: [C![-1,0,1],C![1,-1,0],C![1,0,1],C![1,1,0]]; // minimal model
 
magma: [C![-1,0,1],C![1,-1/2,0],C![1,0,1],C![1,1/2,0]]; // simplified model
 

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: \(\Z/{12}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((1 : 0 : 1) - (1 : -1 : 0)\) \(z (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(x^3 - z^3\) \(0\) \(12\)
Generator $D_0$ Height Order
\((1 : 0 : 1) - (1 : -1 : 0)\) \(z (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(x^3 - z^3\) \(0\) \(12\)
Generator $D_0$ Height Order
\((1 : 0 : 1) - (1 : -1/2 : 0)\) \(z (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(1/2x^3 - 1/2z^3\) \(0\) \(12\)

2-torsion field: 4.0.432.1

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(1\)
Regulator: \( 1 \)
Real period: \( 9.071482 \)
Tamagawa product: \( 6 \)
Torsion order:\( 12 \)
Leading coefficient: \( 0.377978 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

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

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $N(\mathrm{U}(1)\times\mathrm{SU}(2))$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{U}(1)\times\mathrm{SU}(2)\)

Decomposition of the Jacobian

Splits over \(\Q\)

Decomposes up to isogeny as the product of the non-isogenous elliptic curves:
  Elliptic curve 24.a5
  Elliptic curve 36.a4

Endomorphisms of the Jacobian

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

Endomorphism ring over \(\Q\):

\(\End (J_{})\)\(\simeq\)an order of index \(2\) in \(\Z \times \Z\)
\(\End (J_{}) \otimes \Q \)\(\simeq\)\(\Q\) \(\times\) \(\Q\)
\(\End (J_{}) \otimes \R\)\(\simeq\) \(\R \times \R\)

Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) \(\Q(\sqrt{-3}) \) with defining polynomial \(x^{2} - x + 1\)

Not of \(\GL_2\)-type over \(\overline{\Q}\)

Endomorphism ring over \(\overline{\Q}\):

\(\End (J_{\overline{\Q}})\)\(\simeq\)an order of index \(4\) in \(\Z \times \Z [\frac{1 + \sqrt{-3}}{2}]\)
\(\End (J_{\overline{\Q}}) \otimes \Q \)\(\simeq\)\(\Q\) \(\times\) \(\Q(\sqrt{-3}) \)
\(\End (J_{\overline{\Q}}) \otimes \R\)\(\simeq\) \(\R \times \C\)