Properties

Label 100036.a.200072.1
Conductor 100036
Discriminant 200072
Mordell-Weil group \(\Z \times \Z\)
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

$y^2 + (x^3 + x^2 + 1)y = x^5 - 8x^3 - x^2 + 12x - 6$ (homogenize, simplify)
$y^2 + (x^3 + x^2z + z^3)y = x^5z - 8x^3z^3 - x^2z^4 + 12xz^5 - 6z^6$ (dehomogenize, simplify)
$y^2 = x^6 + 6x^5 + x^4 - 30x^3 - 2x^2 + 48x - 23$ (minimize, homogenize)

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-6, 12, -1, -8, 0, 1], R![1, 0, 1, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-6, 12, -1, -8, 0, 1]), R([1, 0, 1, 1]));
 
magma: X,pi:= SimplifiedModel(C);
 
sage: X = HyperellipticCurve(R([-23, 48, -2, -30, 1, 6, 1]))
 

Invariants

Conductor: \( N \)  =  \(100036\) = \( 2^{2} \cdot 89 \cdot 281 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(200072\) = \( 2^{3} \cdot 89 \cdot 281 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(22472\) =  \( 2^{3} \cdot 53^{2} \)
\( I_4 \)  = \(346180\) =  \( 2^{2} \cdot 5 \cdot 19 \cdot 911 \)
\( I_6 \)  = \(2597903816\) =  \( 2^{3} \cdot 1019 \cdot 318683 \)
\( I_{10} \)  = \(819494912\) =  \( 2^{15} \cdot 89 \cdot 281 \)
\( J_2 \)  = \(2809\) =  \( 53^{2} \)
\( J_4 \)  = \(325164\) =  \( 2^{2} \cdot 3 \cdot 7^{3} \cdot 79 \)
\( J_6 \)  = \(49609856\) =  \( 2^{7} \cdot 387577 \)
\( J_8 \)  = \(8405614652\) =  \( 2^{2} \cdot 2101403663 \)
\( J_{10} \)  = \(200072\) =  \( 2^{3} \cdot 89 \cdot 281 \)
\( g_1 \)  = \(174887470365513049/200072\)
\( g_2 \)  = \(1801763080537539/50018\)
\( g_3 \)  = \(48930703272592/25009\)

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
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((1 : -1 : 1)\) \((-2 : 1 : 1)\) \((1 : -2 : 1)\) \((-2 : 2 : 1)\)
\((4 : -61 : 3)\) \((4 : -78 : 3)\)

magma: [C![-2,1,1],C![-2,2,1],C![1,-2,1],C![1,-1,0],C![1,-1,1],C![1,0,0],C![4,-78,3],C![4,-61,3]];
 

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: \(\Z \times \Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((-2 : 2 : 1) + (1 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \((x - z) (x + 2z)\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2\) \(0.301384\) \(\infty\)
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 + xz - 3z^2\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2 - z^3\) \(0.100952\) \(\infty\)

2-torsion field: 6.6.12804608.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(2\)
Mordell-Weil rank: \(2\)
2-Selmer rank:\(2\)
Regulator: \( 0.028791 \)
Real period: \( 14.12244 \)
Tamagawa product: \( 3 \)
Torsion order:\( 1 \)
Leading coefficient: \( 1.219825 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(2\) \(3\) \(2\) \(3\) \(1 + T + T^{2}\)
\(89\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 - 14 T + 89 T^{2} )\)
\(281\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 - 20 T + 281 T^{2} )\)

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 \(\Q\).