Properties

Label 100035.b.300105.1
Conductor 100035
Discriminant 300105
Mordell-Weil group \(\Z \times \Z/{2}\Z\)
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

Related objects

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Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^3 + 1)y = x^5 - 3x^4 - 5x^3 + 11x^2 - 5$ (homogenize, simplify)
$y^2 + (x^3 + z^3)y = x^5z - 3x^4z^2 - 5x^3z^3 + 11x^2z^4 - 5z^6$ (dehomogenize, simplify)
$y^2 = x^6 + 4x^5 - 12x^4 - 18x^3 + 44x^2 - 19$ (minimize, homogenize)

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-5, 0, 11, -5, -3, 1], R![1, 0, 0, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-5, 0, 11, -5, -3, 1]), R([1, 0, 0, 1]));
 
magma: X,pi:= SimplifiedModel(C);
 
sage: X = HyperellipticCurve(R([-19, 0, 44, -18, -12, 4, 1]))
 

Invariants

Conductor: \( N \)  =  \(100035\) = \( 3^{4} \cdot 5 \cdot 13 \cdot 19 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(300105\) = \( 3^{5} \cdot 5 \cdot 13 \cdot 19 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(14952\) =  \( 2^{3} \cdot 3 \cdot 7 \cdot 89 \)
\( I_4 \)  = \(928836\) =  \( 2^{2} \cdot 3^{2} \cdot 25801 \)
\( I_6 \)  = \(4462396488\) =  \( 2^{3} \cdot 3^{3} \cdot 11 \cdot 241 \cdot 7793 \)
\( I_{10} \)  = \(1229230080\) =  \( 2^{12} \cdot 3^{5} \cdot 5 \cdot 13 \cdot 19 \)
\( J_2 \)  = \(1869\) =  \( 3 \cdot 7 \cdot 89 \)
\( J_4 \)  = \(135873\) =  \( 3^{2} \cdot 31 \cdot 487 \)
\( J_6 \)  = \(12388689\) =  \( 3^{2} \cdot 911 \cdot 1511 \)
\( J_8 \)  = \(1173246903\) =  \( 3^{3} \cdot 19 \cdot 2287031 \)
\( J_{10} \)  = \(300105\) =  \( 3^{5} \cdot 5 \cdot 13 \cdot 19 \)
\( g_1 \)  = \(93851287159343/1235\)
\( g_2 \)  = \(3650520528599/1235\)
\( g_3 \)  = \(534267719209/3705\)

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

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

magma: [C![1,-1,0],C![1,-1,1],C![1,0,0]];
 

Number of rational Weierstrass points: \(1\)

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

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((1 : -1 : 1) - (1 : 0 : 0)\) \(z (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(-x^3\) \(0.246223\) \(\infty\)
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 - xz - z^2\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2 - z^3\) \(0\) \(2\)

2-torsion field: 6.6.5559445125.3

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(2\)
Regulator: \( 0.246223 \)
Real period: \( 15.99319 \)
Tamagawa product: \( 2 \)
Torsion order:\( 2 \)
Leading coefficient: \( 1.968946 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(3\) \(5\) \(4\) \(2\) \(1 + T\)
\(5\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + T + 5 T^{2} )\)
\(13\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 - 2 T + 13 T^{2} )\)
\(19\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 4 T + 19 T^{2} )\)

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 \(\Q\).