Properties

Label 100293.a.902637.1
Conductor 100293
Discriminant -902637
Mordell-Weil group \(\Z/{3}\Z\)
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

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-2, 2, -7, 3, -5, 1, -1], R![1, 1, 0, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-2, 2, -7, 3, -5, 1, -1]), R([1, 1, 0, 1]))
 

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-2, 2, -7, 3, -5, 1, -1], R![1, 1, 0, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-7, 10, -27, 14, -18, 4, -3]))
 

$y^2 + (x^3 + x + 1)y = -x^6 + x^5 - 5x^4 + 3x^3 - 7x^2 + 2x - 2$ (homogenize, simplify)
$y^2 + (x^3 + xz^2 + z^3)y = -x^6 + x^5z - 5x^4z^2 + 3x^3z^3 - 7x^2z^4 + 2xz^5 - 2z^6$ (dehomogenize, simplify)
$y^2 = -3x^6 + 4x^5 - 18x^4 + 14x^3 - 27x^2 + 10x - 7$ (minimize, homogenize)

Invariants

\( N \)  =  \(100293\) = \( 3 \cdot 101 \cdot 331 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
\( \Delta \)  =  \(-902637\) = \( - 3^{3} \cdot 101 \cdot 331 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

magma: IgusaClebschInvariants(C); [Factorization(Integers()!a): a in $1];
 
sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]
 

Igusa invariants

magma: IgusaInvariants(C); [Factorization(Integers()!a): a in $1];
 

G2 invariants

magma: G2Invariants(C);
 

\( I_2 \)  = \(-10040\) =  \( - 2^{3} \cdot 5 \cdot 251 \)
\( I_4 \)  = \(602980\) =  \( 2^{2} \cdot 5 \cdot 7 \cdot 59 \cdot 73 \)
\( I_6 \)  = \(-1884167064\) =  \( - 2^{3} \cdot 3^{2} \cdot 13 \cdot 79 \cdot 83 \cdot 307 \)
\( I_{10} \)  = \(-3697201152\) =  \( - 2^{12} \cdot 3^{3} \cdot 101 \cdot 331 \)
\( J_2 \)  = \(-1255\) =  \( - 5 \cdot 251 \)
\( J_4 \)  = \(59345\) =  \( 5 \cdot 11 \cdot 13 \cdot 83 \)
\( J_6 \)  = \(-3494111\) =  \( - 1223 \cdot 2857 \)
\( J_8 \)  = \(215820070\) =  \( 2 \cdot 5 \cdot 107 \cdot 201701 \)
\( J_{10} \)  = \(-902637\) =  \( - 3^{3} \cdot 101 \cdot 331 \)
\( g_1 \)  = \(3113283207034375/902637\)
\( g_2 \)  = \(117304672574375/902637\)
\( g_3 \)  = \(5503312177775/902637\)

Automorphism group

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

Rational points

magma: [];
 

This curve has no rational points.

magma: #Roots(HyperellipticPolynomials(SimplifiedModel(C)));
 

Number of rational Weierstrass points: \(0\)

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);
 

This curve is locally solvable except over $\R$.

Mordell-Weil group of the Jacobian:

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Group structure: \(\Z/{3}\Z\)

Generator Height Order
\(x^2 + 2z^2\) \(=\) \(0,\) \(y\) \(=\) \(xz^2 - z^3\) \(0\) \(3\)

2-torsion field: 6.0.6418752.1

BSD invariants

Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(1\)
Regulator: \( 1 \)
Real period: \( 4.422795 \)
Tamagawa product: \( 3 \)
Torsion order:\( 3 \)
Leading coefficient: \( 2.948530 \)
Analytic order of Ш: \( 2 \)   (rounded)
Order of Ш:twice a square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(3\) \(3\) \(1\) \(3\) \(( 1 - T )( 1 + T + 3 T^{2} )\)
\(101\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 101 T^{2} )\)
\(331\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 - 5 T + 331 T^{2} )\)

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 \(\Q\).