Properties

Label 1239.a.8673.1
Conductor 1239
Discriminant 8673
Mordell-Weil group \(\Z/{6}\Z\)
Sato-Tate group $\mathrm{USp}(4)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type no

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Minimal equation

Minimal equation

Simplified equation

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

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -1, -1, 0, 0, 0, -1]), R([1, 1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -1, -1, 0, 0, 0, -1], R![1, 1, 1]);
 
sage: X = HyperellipticCurve(R([1, -2, -1, 2, 1, 0, -4]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(1239\) \(=\) \( 3 \cdot 7 \cdot 59 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(8673\) \(=\) \( 3 \cdot 7^{2} \cdot 59 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(1000\) \(=\)  \( 2^{3} \cdot 5^{3} \)
\( I_4 \)  \(=\) \(17092\) \(=\)  \( 2^{2} \cdot 4273 \)
\( I_6 \)  \(=\) \(5030856\) \(=\)  \( 2^{3} \cdot 3^{3} \cdot 23291 \)
\( I_{10} \)  \(=\) \(35524608\) \(=\)  \( 2^{12} \cdot 3 \cdot 7^{2} \cdot 59 \)
\( J_2 \)  \(=\) \(125\) \(=\)  \( 5^{3} \)
\( J_4 \)  \(=\) \(473\) \(=\)  \( 11 \cdot 43 \)
\( J_6 \)  \(=\) \(1969\) \(=\)  \( 11 \cdot 179 \)
\( J_8 \)  \(=\) \(5599\) \(=\)  \( 11 \cdot 509 \)
\( J_{10} \)  \(=\) \(8673\) \(=\)  \( 3 \cdot 7^{2} \cdot 59 \)
\( g_1 \)  \(=\) \(30517578125/8673\)
\( g_2 \)  \(=\) \(923828125/8673\)
\( g_3 \)  \(=\) \(30765625/8673\)

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: \((0 : 0 : 1),\, (0 : -1 : 1),\, (1 : -7 : 2)\)

magma: [C![0,-1,1],C![0,0,1],C![1,-7,2]];
 

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

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((0 : 0 : 1) + (1 : -7 : 2) - D_\infty\) \((-2x + z) x\) \(=\) \(0,\) \(4y\) \(=\) \(-7xz^2\) \(0\) \(6\)

2-torsion field: 6.0.93987.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(1\)
Regulator: \( 1 \)
Real period: \( 8.456792 \)
Tamagawa product: \( 2 \)
Torsion order:\( 6 \)
Leading coefficient: \( 0.469821 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(3\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 - T + 3 T^{2} )\)
\(7\) \(1\) \(2\) \(2\) \(( 1 + T )( 1 + T + 7 T^{2} )\)
\(59\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 59 T^{2} )\)

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 \(\Q\).