Properties

Label 64237.a.64237.1
Conductor $64237$
Discriminant $64237$
Mordell-Weil group \(\Z \oplus \Z \oplus \Z\)
Sato-Tate group $\mathrm{USp}(4)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q\)
\(\End(J) \otimes \Q\) \(\Q\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type no

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Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^3 + x^2 + 1)y = x^3 - 3x$ (homogenize, simplify)
$y^2 + (x^3 + x^2z + z^3)y = x^3z^3 - 3xz^5$ (dehomogenize, simplify)
$y^2 = x^6 + 2x^5 + x^4 + 6x^3 + 2x^2 - 12x + 1$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -3, 0, 1]), R([1, 0, 1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -3, 0, 1], R![1, 0, 1, 1]);
 
sage: X = HyperellipticCurve(R([1, -12, 2, 6, 1, 2, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(64237\) \(=\) \( 64237 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(64237\) \(=\) \( 64237 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(508\) \(=\)  \( 2^{2} \cdot 127 \)
\( I_4 \)  \(=\) \(4489\) \(=\)  \( 67^{2} \)
\( I_6 \)  \(=\) \(4345419\) \(=\)  \( 3 \cdot 13 \cdot 67 \cdot 1663 \)
\( I_{10} \)  \(=\) \(-8222336\) \(=\)  \( - 2^{7} \cdot 64237 \)
\( J_2 \)  \(=\) \(127\) \(=\)  \( 127 \)
\( J_4 \)  \(=\) \(485\) \(=\)  \( 5 \cdot 97 \)
\( J_6 \)  \(=\) \(-49013\) \(=\)  \( - 23 \cdot 2131 \)
\( J_8 \)  \(=\) \(-1614969\) \(=\)  \( - 3^{2} \cdot 179441 \)
\( J_{10} \)  \(=\) \(-64237\) \(=\)  \( -64237 \)
\( g_1 \)  \(=\) \(-33038369407/64237\)
\( g_2 \)  \(=\) \(-993465755/64237\)
\( g_3 \)  \(=\) \(790530677/64237\)

sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]
 
magma: IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);
 

Automorphism group

\(\mathrm{Aut}(X)\)\(\simeq\) $C_2$
magma: AutomorphismGroup(C); IdentifyGroup($1);
 
\(\mathrm{Aut}(X_{\overline{\Q}})\)\(\simeq\) $C_2$
magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
 

Rational points

Known points
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 0 : 1)\) \((0 : -1 : 1)\) \((-1 : 1 : 1)\) \((1 : -1 : 1)\)
\((-2 : 1 : 1)\) \((-1 : -2 : 1)\) \((1 : -2 : 1)\) \((-2 : 2 : 1)\) \((-3 : -8 : 2)\) \((-3 : 9 : 2)\)
\((-2 : 23 : 3)\) \((25 : 23 : 3)\) \((-2 : -54 : 3)\) \((-23 : 1472 : 12)\) \((-23 : 2619 : 12)\) \((25 : -17550 : 3)\)
Known points
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 0 : 1)\) \((0 : -1 : 1)\) \((-1 : 1 : 1)\) \((1 : -1 : 1)\)
\((-2 : 1 : 1)\) \((-1 : -2 : 1)\) \((1 : -2 : 1)\) \((-2 : 2 : 1)\) \((-3 : -8 : 2)\) \((-3 : 9 : 2)\)
\((-2 : 23 : 3)\) \((25 : 23 : 3)\) \((-2 : -54 : 3)\) \((-23 : 1472 : 12)\) \((-23 : 2619 : 12)\) \((25 : -17550 : 3)\)
Known points
\((1 : -1 : 0)\) \((1 : 1 : 0)\) \((0 : -1 : 1)\) \((0 : 1 : 1)\) \((1 : -1 : 1)\) \((1 : 1 : 1)\)
\((-2 : -1 : 1)\) \((-2 : 1 : 1)\) \((-1 : -3 : 1)\) \((-1 : 3 : 1)\) \((-3 : -17 : 2)\) \((-3 : 17 : 2)\)
\((-2 : -77 : 3)\) \((-2 : 77 : 3)\) \((-23 : -1147 : 12)\) \((-23 : 1147 : 12)\) \((25 : -17573 : 3)\) \((25 : 17573 : 3)\)

magma: [C![-23,1472,12],C![-23,2619,12],C![-3,-8,2],C![-3,9,2],C![-2,-54,3],C![-2,1,1],C![-2,2,1],C![-2,23,3],C![-1,-2,1],C![-1,1,1],C![0,-1,1],C![0,0,1],C![1,-2,1],C![1,-1,0],C![1,-1,1],C![1,0,0],C![25,-17550,3],C![25,23,3]]; // minimal model
 
magma: [C![-23,-1147,12],C![-23,1147,12],C![-3,-17,2],C![-3,17,2],C![-2,-77,3],C![-2,-1,1],C![-2,1,1],C![-2,77,3],C![-1,-3,1],C![-1,3,1],C![0,-1,1],C![0,1,1],C![1,-1,1],C![1,-1,0],C![1,1,1],C![1,1,0],C![25,-17573,3],C![25,17573,3]]; // simplified model
 

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: \(\Z \oplus \Z \oplus \Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((-2 : 2 : 1) + (1 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \((x - z) (x + 2z)\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2\) \(0.428326\) \(\infty\)
\((-2 : 1 : 1) - (1 : -1 : 0)\) \(z (x + 2z)\) \(=\) \(0,\) \(y\) \(=\) \(z^3\) \(0.392254\) \(\infty\)
\(D_0 - 2 \cdot(1 : -1 : 0)\) \(z^2\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0.271869\) \(\infty\)
Generator $D_0$ Height Order
\((-2 : 2 : 1) + (1 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \((x - z) (x + 2z)\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2\) \(0.428326\) \(\infty\)
\((-2 : 1 : 1) - (1 : -1 : 0)\) \(z (x + 2z)\) \(=\) \(0,\) \(y\) \(=\) \(z^3\) \(0.392254\) \(\infty\)
\(D_0 - 2 \cdot(1 : -1 : 0)\) \(z^2\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0.271869\) \(\infty\)
Generator $D_0$ Height Order
\((-2 : 1 : 1) + (1 : 1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) \((x - z) (x + 2z)\) \(=\) \(0,\) \(y\) \(=\) \(x^3 + x^2z - 2xz^2 + z^3\) \(0.428326\) \(\infty\)
\((-2 : -1 : 1) - (1 : -1 : 0)\) \(z (x + 2z)\) \(=\) \(0,\) \(y\) \(=\) \(x^3 + x^2z + 3z^3\) \(0.392254\) \(\infty\)
\(D_0 - 2 \cdot(1 : -1 : 0)\) \(z^2\) \(=\) \(0,\) \(y\) \(=\) \(x^3 + x^2z + z^3\) \(0.271869\) \(\infty\)

2-torsion field: 6.2.4111168.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(3\)   (upper bound)
Mordell-Weil rank: \(3\)
2-Selmer rank:\(3\)
Regulator: \( 0.034994 \)
Real period: \( 18.15399 \)
Tamagawa product: \( 1 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.635284 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(64237\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 - 38 T + 64237 T^{2} )\)

Galois representations

The mod-$\ell$ Galois representation has maximal image \(\GSp(4,\F_\ell)\) for all primes \( \ell \) .

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $\mathrm{USp}(4)$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{USp}(4)\)

Decomposition of the Jacobian

Simple over \(\overline{\Q}\)

magma: HeuristicDecompositionFactors(C);
 

Endomorphisms of the Jacobian

Not of \(\GL_2\)-type over \(\Q\)

Endomorphism ring over \(\Q\):

\(\End (J_{})\)\(\simeq\)\(\Z\)
\(\End (J_{}) \otimes \Q \)\(\simeq\)\(\Q\)
\(\End (J_{}) \otimes \R\)\(\simeq\) \(\R\)

All \(\overline{\Q}\)-endomorphisms of the Jacobian are defined over \(\Q\).

magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 

magma: HeuristicIsGL2(C); HeuristicEndomorphismDescription(C); HeuristicEndomorphismFieldOfDefinition(C);
 

magma: HeuristicIsGL2(C : Geometric := true); HeuristicEndomorphismDescription(C : Geometric := true); HeuristicEndomorphismLatticeDescription(C);