Properties

Label 46234.a.92468.1
Conductor 46234
Discriminant 92468
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

$y^2 + (x^3 + 1)y = -x^4 + 3x^2 - 2x$ (homogenize, simplify)
$y^2 + (x^3 + z^3)y = -x^4z^2 + 3x^2z^4 - 2xz^5$ (dehomogenize, simplify)
$y^2 = x^6 - 4x^4 + 2x^3 + 12x^2 - 8x + 1$ (minimize, homogenize)

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -2, 3, 0, -1], R![1, 0, 0, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -2, 3, 0, -1]), R([1, 0, 0, 1]));
 
magma: X,pi:= SimplifiedModel(C);
 
sage: X = HyperellipticCurve(R([1, -8, 12, 2, -4, 0, 1]))
 

Invariants

Conductor: \( N \)  =  \(46234\) = \( 2 \cdot 23117 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(92468\) = \( 2^{2} \cdot 23117 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(552\) =  \( 2^{3} \cdot 3 \cdot 23 \)
\( I_4 \)  = \(77028\) =  \( 2^{2} \cdot 3 \cdot 7^{2} \cdot 131 \)
\( I_6 \)  = \(6317160\) =  \( 2^{3} \cdot 3 \cdot 5 \cdot 61 \cdot 863 \)
\( I_{10} \)  = \(378748928\) =  \( 2^{14} \cdot 23117 \)
\( J_2 \)  = \(69\) =  \( 3 \cdot 23 \)
\( J_4 \)  = \(-604\) =  \( - 2^{2} \cdot 151 \)
\( J_6 \)  = \(5172\) =  \( 2^{2} \cdot 3 \cdot 431 \)
\( J_8 \)  = \(-1987\) =  \( - 1987 \)
\( J_{10} \)  = \(92468\) =  \( 2^{2} \cdot 23117 \)
\( g_1 \)  = \(1564031349/92468\)
\( g_2 \)  = \(-49604859/23117\)
\( g_3 \)  = \(6155973/23117\)

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
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 0 : 1)\) \((0 : -1 : 1)\) \((1 : 0 : 1)\) \((-2 : 0 : 1)\)
\((-1 : -2 : 1)\) \((-1 : 2 : 1)\) \((1 : -2 : 1)\) \((2 : -1 : 1)\) \((-3 : 2 : 1)\) \((1 : -4 : 2)\)
\((1 : -5 : 2)\) \((-2 : 7 : 1)\) \((2 : -8 : 1)\) \((-3 : 24 : 1)\) \((1 : -100 : 6)\) \((1 : -117 : 6)\)
\((7 : -496 : 12)\) \((7 : -1575 : 12)\)

magma: [C![-3,2,1],C![-3,24,1],C![-2,0,1],C![-2,7,1],C![-1,-2,1],C![-1,2,1],C![0,-1,1],C![0,0,1],C![1,-117,6],C![1,-100,6],C![1,-5,2],C![1,-4,2],C![1,-2,1],C![1,-1,0],C![1,0,0],C![1,0,1],C![2,-8,1],C![2,-1,1],C![7,-1575,12],C![7,-496,12]];
 

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
\((1 : -2 : 1) - (1 : -1 : 0)\) \(z (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(-2z^3\) \(0.353477\) \(\infty\)
\((0 : 0 : 1) + (1 : -2 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \((-x + z) x\) \(=\) \(0,\) \(y\) \(=\) \(-2xz^2\) \(0.292010\) \(\infty\)
\((0 : -1 : 1) - (1 : 0 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 - z^3\) \(0.209095\) \(\infty\)

2-torsion field: 6.2.1479488.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(3\)   (upper bound)
Mordell-Weil rank: \(3\)
2-Selmer rank:\(3\)
Regulator: \( 0.015896 \)
Real period: \( 19.58611 \)
Tamagawa product: \( 2 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.622688 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(2\) \(2\) \(1\) \(2\) \(( 1 + T )( 1 + T + 2 T^{2} )\)
\(23117\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 200 T + 23117 T^{2} )\)

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 \(\Q\).