Properties

Label 100278.a.200556.1
Conductor 100278
Discriminant -200556
Mordell-Weil group \(\Z\)
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

$y^2 + x^2y = 2x^5 + 9x^4 + x^3 - 21x^2 + 15x - 3$ (homogenize, simplify)
$y^2 + x^2zy = 2x^5z + 9x^4z^2 + x^3z^3 - 21x^2z^4 + 15xz^5 - 3z^6$ (dehomogenize, simplify)
$y^2 = 8x^5 + 37x^4 + 4x^3 - 84x^2 + 60x - 12$ (minimize, homogenize)

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-3, 15, -21, 1, 9, 2], R![0, 0, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-3, 15, -21, 1, 9, 2]), R([0, 0, 1]));
 
magma: X,pi:= SimplifiedModel(C);
 
sage: X = HyperellipticCurve(R([-12, 60, -84, 4, 37, 8]))
 

Invariants

Conductor: \( N \)  =  \(100278\) = \( 2 \cdot 3^{4} \cdot 619 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(-200556\) = \( - 2^{2} \cdot 3^{4} \cdot 619 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(69024\) =  \( 2^{5} \cdot 3 \cdot 719 \)
\( I_4 \)  = \(598464\) =  \( 2^{6} \cdot 3^{2} \cdot 1039 \)
\( I_6 \)  = \(14277219264\) =  \( 2^{6} \cdot 3^{2} \cdot 7 \cdot 11 \cdot 487 \cdot 661 \)
\( I_{10} \)  = \(-821477376\) =  \( - 2^{14} \cdot 3^{4} \cdot 619 \)
\( J_2 \)  = \(8628\) =  \( 2^{2} \cdot 3 \cdot 719 \)
\( J_4 \)  = \(3095532\) =  \( 2^{2} \cdot 3^{2} \cdot 11 \cdot 7817 \)
\( J_6 \)  = \(1476933817\) =  \( 379 \cdot 3896923 \)
\( J_8 \)  = \(790166652513\) =  \( 3 \cdot 7 \cdot 37626983453 \)
\( J_{10} \)  = \(-200556\) =  \( - 2^{2} \cdot 3^{4} \cdot 619 \)
\( g_1 \)  = \(-147572580633292032/619\)
\( g_2 \)  = \(-6136499412390336/619\)
\( g_3 \)  = \(-3054068731880548/5571\)

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: \((1 : 0 : 0),\, (1 : 0 : 2),\, (1 : -2 : 2)\)

magma: [C![1,-2,2],C![1,0,0],C![1,0,2]];
 

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: \(\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(2x^2 + 5xz - 4z^2\) \(=\) \(0,\) \(y\) \(=\) \(xz^2 - z^3\) \(0.165292\) \(\infty\)

2-torsion field: 5.3.802224.2

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(1\)
Regulator: \( 0.165292 \)
Real period: \( 7.453400 \)
Tamagawa product: \( 2 \)
Torsion order:\( 1 \)
Leading coefficient: \( 2.463979 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(2\) \(2\) \(1\) \(2\) \(( 1 + T )( 1 - 2 T + 2 T^{2} )\)
\(3\) \(4\) \(4\) \(1\) \(1 + 2 T + 3 T^{2}\)
\(619\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 4 T + 619 T^{2} )\)

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 \(\Q\).