Properties

Label 67006.a.134012.1
Conductor 67006
Discriminant 134012
Mordell-Weil group \(\Z \times \Z \times \Z\)
Sato-Tate group $\mathrm{USp}(4)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type no

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Minimal equation

Minimal equation

Simplified equation

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

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 0, -1, 1, 3], R![1, 0, 0, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 0, -1, 1, 3]), R([1, 0, 0, 1]));
 
magma: X,pi:= SimplifiedModel(C);
 
sage: X = HyperellipticCurve(R([1, 0, -4, 6, 12, 0, 1]))
 

Invariants

Conductor: \( N \)  =  \(67006\) = \( 2 \cdot 33503 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(134012\) = \( 2^{2} \cdot 33503 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(744\) =  \( 2^{3} \cdot 3 \cdot 31 \)
\( I_4 \)  = \(126564\) =  \( 2^{2} \cdot 3 \cdot 53 \cdot 199 \)
\( I_6 \)  = \(20053992\) =  \( 2^{3} \cdot 3 \cdot 7 \cdot 79 \cdot 1511 \)
\( I_{10} \)  = \(548913152\) =  \( 2^{14} \cdot 33503 \)
\( J_2 \)  = \(93\) =  \( 3 \cdot 31 \)
\( J_4 \)  = \(-958\) =  \( - 2 \cdot 479 \)
\( J_6 \)  = \(1104\) =  \( 2^{4} \cdot 3 \cdot 23 \)
\( J_8 \)  = \(-203773\) =  \( - 203773 \)
\( J_{10} \)  = \(134012\) =  \( 2^{2} \cdot 33503 \)
\( g_1 \)  = \(6956883693/134012\)
\( g_2 \)  = \(-385287003/67006\)
\( g_3 \)  = \(2387124/33503\)

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

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
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 0 : 1)\) \((0 : -1 : 1)\) \((-1 : -1 : 1)\) \((-1 : 1 : 1)\)
\((1 : 1 : 1)\) \((1 : -3 : 1)\) \((1 : -1 : 3)\) \((-1 : -3 : 2)\) \((-1 : -4 : 2)\) \((2 : 4 : 1)\)
\((2 : -13 : 1)\) \((-3 : -17 : 2)\) \((-2 : -25 : 5)\) \((1 : -27 : 3)\) \((-3 : 36 : 2)\) \((-2 : -92 : 5)\)

magma: [C![-3,-17,2],C![-3,36,2],C![-2,-92,5],C![-2,-25,5],C![-1,-4,2],C![-1,-3,2],C![-1,-1,1],C![-1,1,1],C![0,-1,1],C![0,0,1],C![1,-27,3],C![1,-3,1],C![1,-1,0],C![1,-1,3],C![1,0,0],C![1,1,1],C![2,-13,1],C![2,4,1]];
 

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: \(\Z \times \Z \times \Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((-1 : -4 : 2) - (1 : -1 : 0)\) \(z (2x + z)\) \(=\) \(0,\) \(2y\) \(=\) \(-z^3\) \(0.720021\) \(\infty\)
\((-1 : -1 : 1) - (1 : -1 : 0)\) \(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.327330\) \(\infty\)
\((-1 : -1 : 1) + (0 : 0 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \(x (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(xz^2\) \(0.085482\) \(\infty\)

2-torsion field: 6.2.134012.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(3\)   (upper bound)
Mordell-Weil rank: \(3\)
2-Selmer rank:\(3\)
Regulator: \( 0.019103 \)
Real period: \( 19.35184 \)
Tamagawa product: \( 2 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.739388 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(2\) \(2\) \(1\) \(2\) \(( 1 + T )( 1 + T + 2 T^{2} )\)
\(33503\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 - 220 T + 33503 T^{2} )\)

Sato-Tate group

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

Decomposition of the Jacobian

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

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 \(\Q\).