Properties

Label 1038.a.1038.1
Conductor $1038$
Discriminant $1038$
Mordell-Weil group \(\Z/{6}\Z\)
Sato-Tate group $\mathrm{USp}(4)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q\)
\(\End(J) \otimes \Q\) \(\Q\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type no

Related objects

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Show commands: SageMath / Magma

Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^2 + x)y = x^5 - 12x^4 + 26x^3 + 46x^2 + 21x + 3$ (homogenize, simplify)
$y^2 + (x^2z + xz^2)y = x^5z - 12x^4z^2 + 26x^3z^3 + 46x^2z^4 + 21xz^5 + 3z^6$ (dehomogenize, simplify)
$y^2 = 4x^5 - 47x^4 + 106x^3 + 185x^2 + 84x + 12$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([3, 21, 46, 26, -12, 1]), R([0, 1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![3, 21, 46, 26, -12, 1], R![0, 1, 1]);
 
sage: X = HyperellipticCurve(R([12, 84, 185, 106, -47, 4]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(1038\) \(=\) \( 2 \cdot 3 \cdot 173 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(1038\) \(=\) \( 2 \cdot 3 \cdot 173 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(109988\) \(=\)  \( 2^{2} \cdot 31 \cdot 887 \)
\( I_4 \)  \(=\) \(334849\) \(=\)  \( 17 \cdot 19697 \)
\( I_6 \)  \(=\) \(12332566337\) \(=\)  \( 7 \cdot 13 \cdot 135522707 \)
\( I_{10} \)  \(=\) \(132864\) \(=\)  \( 2^{8} \cdot 3 \cdot 173 \)
\( J_2 \)  \(=\) \(27497\) \(=\)  \( 31 \cdot 887 \)
\( J_4 \)  \(=\) \(31489590\) \(=\)  \( 2 \cdot 3 \cdot 5 \cdot 11 \cdot 37 \cdot 2579 \)
\( J_6 \)  \(=\) \(48060441688\) \(=\)  \( 2^{3} \cdot 7 \cdot 269 \cdot 3190417 \)
\( J_8 \)  \(=\) \(82480921681709\) \(=\)  \( 82480921681709 \)
\( J_{10} \)  \(=\) \(1038\) \(=\)  \( 2 \cdot 3 \cdot 173 \)
\( g_1 \)  \(=\) \(15719059879327073637257/1038\)
\( g_2 \)  \(=\) \(109111794064913809345/173\)
\( g_3 \)  \(=\) \(18168889743107727596/519\)

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

magma: [C![1,0,0]]; // minimal model
 
magma: [C![1,0,0]]; // simplified model
 

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: \(\Z/{6}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 - 4xz - 2z^2\) \(=\) \(0,\) \(y\) \(=\) \(-7xz^2 - 3z^3\) \(0\) \(6\)
Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 - 4xz - 2z^2\) \(=\) \(0,\) \(y\) \(=\) \(-7xz^2 - 3z^3\) \(0\) \(6\)
Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 - 4xz - 2z^2\) \(=\) \(0,\) \(y\) \(=\) \(x^2z - 13xz^2 - 6z^3\) \(0\) \(6\)

2-torsion field: 6.6.827476992.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(1\)
Regulator: \( 1 \)
Real period: \( 15.39734 \)
Tamagawa product: \( 1 \)
Torsion order:\( 6 \)
Leading coefficient: \( 0.427704 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 - T + 2 T^{2} )\)
\(3\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 - T + 3 T^{2} )\)
\(173\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 - 6 T + 173 T^{2} )\)

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 \(\Q\).