Properties

Label 100154.b.200308.1
Conductor 100154
Discriminant 200308
Mordell-Weil group \(\Z/{2}\Z\)
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

$y^2 + (x + 1)y = x^5 + x^3 - 5x^2 + 3x - 1$ (homogenize, simplify)
$y^2 + (xz^2 + z^3)y = x^5z + x^3z^3 - 5x^2z^4 + 3xz^5 - z^6$ (dehomogenize, simplify)
$y^2 = 4x^5 + 4x^3 - 19x^2 + 14x - 3$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-1, 3, -5, 1, 0, 1]), R([1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-1, 3, -5, 1, 0, 1], R![1, 1]);
 
sage: X = HyperellipticCurve(R([-3, 14, -19, 4, 0, 4]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  =  \(100154\) = \( 2 \cdot 50077 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(200308\) = \( 2^{2} \cdot 50077 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(2336\) =  \( 2^{5} \cdot 73 \)
\( I_4 \)  = \(-14336\) =  \( - 2^{11} \cdot 7 \)
\( I_6 \)  = \(-23492416\) =  \( - 2^{6} \cdot 367069 \)
\( I_{10} \)  = \(820461568\) =  \( 2^{14} \cdot 50077 \)
\( J_2 \)  = \(292\) =  \( 2^{2} \cdot 73 \)
\( J_4 \)  = \(3702\) =  \( 2 \cdot 3 \cdot 617 \)
\( J_6 \)  = \(86305\) =  \( 5 \cdot 41 \cdot 421 \)
\( J_8 \)  = \(2874064\) =  \( 2^{4} \cdot 263 \cdot 683 \)
\( J_{10} \)  = \(200308\) =  \( 2^{2} \cdot 50077 \)
\( g_1 \)  = \(530706327808/50077\)
\( g_2 \)  = \(23042254944/50077\)
\( g_3 \)  = \(1839677380/50077\)

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

All points: \((1 : 0 : 0),\, (1 : -1 : 1)\)

magma: [C![1,-1,1],C![1,0,0]];
 

Number of rational Weierstrass points: \(2\)

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: \(\Z/{2}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

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

2-torsion field: 4.0.200308.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(1\)
Regulator: \( 1 \)
Real period: \( 4.011604 \)
Tamagawa product: \( 2 \)
Torsion order:\( 2 \)
Leading coefficient: \( 2.005802 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(2\) \(2\) \(1\) \(2\) \(( 1 - T )( 1 + 2 T^{2} )\)
\(50077\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 - 236 T + 50077 T^{2} )\)

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 \(\Q\).