Properties

Label 644.a.659456.1
Conductor 644
Discriminant 659456
Mordell-Weil group \(\Z/{2}\Z\)
Sato-Tate group $G_{3,3}$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R \times \R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q \times \Q\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type yes

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Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^2 + x)y = -3x^6 - 13x^5 + 4x^4 + 51x^3 + 4x^2 - 13x - 3$ (homogenize, simplify)
$y^2 + (x^2z + xz^2)y = -3x^6 - 13x^5z + 4x^4z^2 + 51x^3z^3 + 4x^2z^4 - 13xz^5 - 3z^6$ (dehomogenize, simplify)
$y^2 = -12x^6 - 52x^5 + 17x^4 + 206x^3 + 17x^2 - 52x - 12$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-3, -13, 4, 51, 4, -13, -3]), R([0, 1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-3, -13, 4, 51, 4, -13, -3], R![0, 1, 1]);
 
sage: X = HyperellipticCurve(R([-12, -52, 17, 206, 17, -52, -12]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  =  \(644\) = \( 2^{2} \cdot 7 \cdot 23 \)
magma: Conductor(LSeries(C: ExcFactors:=[*<2,Valuation(644,2),R![1, 2, 1]>*])); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(659456\) = \( 2^{12} \cdot 7 \cdot 23 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(323592\) =  \( 2^{3} \cdot 3 \cdot 97 \cdot 139 \)
\( I_4 \)  = \(4282649220\) =  \( 2^{2} \cdot 3 \cdot 5 \cdot 23 \cdot 1487 \cdot 2087 \)
\( I_6 \)  = \(368522127355272\) =  \( 2^{3} \cdot 3 \cdot 11 \cdot 1395917149073 \)
\( I_{10} \)  = \(2701131776\) =  \( 2^{24} \cdot 7 \cdot 23 \)
\( J_2 \)  = \(40449\) =  \( 3 \cdot 97 \cdot 139 \)
\( J_4 \)  = \(23560804\) =  \( 2^{2} \cdot 199 \cdot 29599 \)
\( J_6 \)  = \(14638854160\) =  \( 2^{4} \cdot 5 \cdot 7 \cdot 23 \cdot 1136557 \)
\( J_8 \)  = \(9253881697856\) =  \( 2^{6} \cdot 54101 \cdot 2672629 \)
\( J_{10} \)  = \(659456\) =  \( 2^{12} \cdot 7 \cdot 23 \)
\( g_1 \)  = \(108277681088425330677249/659456\)
\( g_2 \)  = \(389810454818831018649/164864\)
\( g_3 \)  = \(9297727292338785/256\)

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: \((-3 : -3 : 1),\, (-1 : 3 : 3)\)

magma: [C![-3,-3,1],C![-1,3,3]];
 

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

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((-3 : -3 : 1) + (-1 : 3 : 3) - D_\infty\) \((x + 3z) (3x + z)\) \(=\) \(0,\) \(6y\) \(=\) \(7xz^2 + 3z^3\) \(0\) \(2\)

2-torsion field: 4.4.10304.1

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(1\)
Regulator: \( 1 \)
Real period: \( 0.872984 \)
Tamagawa product: \( 1 \)
Torsion order:\( 2 \)
Leading coefficient: \( 0.218246 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(2\) \(12\) \(2\) \(1\) \(( 1 + T )^{2}\)
\(7\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 4 T + 7 T^{2} )\)
\(23\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 23 T^{2} )\)

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $G_{3,3}$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{SU}(2)\times\mathrm{SU}(2)\)

Decomposition of the Jacobian

Splits over \(\Q\)

Decomposes up to isogeny as the product of the non-isogenous elliptic curves:
  Elliptic curve 46.a1
  Elliptic curve 14.a1

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 \(\Q\).