Properties

Label 100162.a.100162.1
Conductor 100162
Discriminant -100162
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

$y^2 + (x^3 + 1)y = x^5 - 2x^4 - 9x^3 + 8x^2 + 20x - 20$ (homogenize, simplify)
$y^2 + (x^3 + z^3)y = x^5z - 2x^4z^2 - 9x^3z^3 + 8x^2z^4 + 20xz^5 - 20z^6$ (dehomogenize, simplify)
$y^2 = x^6 + 4x^5 - 8x^4 - 34x^3 + 32x^2 + 80x - 79$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-20, 20, 8, -9, -2, 1]), R([1, 0, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-20, 20, 8, -9, -2, 1], R![1, 0, 0, 1]);
 
sage: X = HyperellipticCurve(R([-79, 80, 32, -34, -8, 4, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  =  \(100162\) = \( 2 \cdot 61 \cdot 821 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(-100162\) = \( - 2 \cdot 61 \cdot 821 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(42792\) =  \( 2^{3} \cdot 3 \cdot 1783 \)
\( I_4 \)  = \(-42012\) =  \( - 2^{2} \cdot 3^{3} \cdot 389 \)
\( I_6 \)  = \(-607587480\) =  \( - 2^{3} \cdot 3^{5} \cdot 5 \cdot 17 \cdot 3677 \)
\( I_{10} \)  = \(-410263552\) =  \( - 2^{13} \cdot 61 \cdot 821 \)
\( J_2 \)  = \(5349\) =  \( 3 \cdot 1783 \)
\( J_4 \)  = \(1192596\) =  \( 2^{2} \cdot 3 \cdot 23 \cdot 29 \cdot 149 \)
\( J_6 \)  = \(354674332\) =  \( 2^{2} \cdot 17 \cdot 5215799 \)
\( J_8 \)  = \(118716945663\) =  \( 3 \cdot 37 \cdot 53 \cdot 3407 \cdot 5923 \)
\( J_{10} \)  = \(-100162\) =  \( - 2 \cdot 61 \cdot 821 \)
\( g_1 \)  = \(-4378879451923801749/100162\)
\( g_2 \)  = \(-91260143303221602/50081\)
\( g_3 \)  = \(-5073935703495966/50081\)

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
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((2 : -4 : 1)\) \((2 : -5 : 1)\) \((4 : -31 : 3)\) \((4 : -60 : 3)\)
\((-13 : 1025 : 4)\) \((-13 : 1108 : 4)\)

magma: [C![-13,1025,4],C![-13,1108,4],C![1,-1,0],C![1,0,0],C![2,-5,1],C![2,-4,1],C![4,-60,3],C![4,-31,3]];
 

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
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 + 2xz - 5z^2\) \(=\) \(0,\) \(y\) \(=\) \(-4xz^2 + 4z^3\) \(0.541687\) \(\infty\)
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 + 2xz - 4z^2\) \(=\) \(0,\) \(y\) \(=\) \(-4xz^2 + 3z^3\) \(0.385982\) \(\infty\)

2-torsion field: 6.4.400648.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(2\)
Mordell-Weil rank: \(2\)
2-Selmer rank:\(2\)
Regulator: \( 0.174345 \)
Real period: \( 6.880813 \)
Tamagawa product: \( 1 \)
Torsion order:\( 1 \)
Leading coefficient: \( 1.199636 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(2\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + T + 2 T^{2} )\)
\(61\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 10 T + 61 T^{2} )\)
\(821\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 821 T^{2} )\)

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 \(\Q\).