Properties

Label 3138.a.301248.1
Conductor 3138
Discriminant -301248
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

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Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^3 + x)y = -x^4 - 2x^2 - x + 1$ (homogenize, simplify)
$y^2 + (x^3 + xz^2)y = -x^4z^2 - 2x^2z^4 - xz^5 + z^6$ (dehomogenize, simplify)
$y^2 = x^6 - 2x^4 - 7x^2 - 4x + 4$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([1, -1, -2, 0, -1]), R([0, 1, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![1, -1, -2, 0, -1], R![0, 1, 0, 1]);
 
sage: X = HyperellipticCurve(R([4, -4, -7, 0, -2, 0, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(3138\) \(=\) \( 2 \cdot 3 \cdot 523 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-301248\) \(=\) \( - 2^{6} \cdot 3^{2} \cdot 523 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(-1184\) \(=\)  \( - 2^{5} \cdot 37 \)
\( I_4 \)  \(=\) \(-29120\) \(=\)  \( - 2^{6} \cdot 5 \cdot 7 \cdot 13 \)
\( I_6 \)  \(=\) \(159680\) \(=\)  \( 2^{6} \cdot 5 \cdot 499 \)
\( I_{10} \)  \(=\) \(-1233911808\) \(=\)  \( - 2^{18} \cdot 3^{2} \cdot 523 \)
\( J_2 \)  \(=\) \(-148\) \(=\)  \( - 2^{2} \cdot 37 \)
\( J_4 \)  \(=\) \(1216\) \(=\)  \( 2^{6} \cdot 19 \)
\( J_6 \)  \(=\) \(4689\) \(=\)  \( 3^{2} \cdot 521 \)
\( J_8 \)  \(=\) \(-543157\) \(=\)  \( -543157 \)
\( J_{10} \)  \(=\) \(-301248\) \(=\)  \( - 2^{6} \cdot 3^{2} \cdot 523 \)
\( g_1 \)  \(=\) \(1109503312/4707\)
\( g_2 \)  \(=\) \(61594048/4707\)
\( g_3 \)  \(=\) \(-713249/2092\)

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
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : -1 : 1)\) \((0 : 1 : 1)\) \((-1 : 1 : 1)\) \((1 : -1 : 2)\)
\((-2 : 3 : 1)\) \((1 : -4 : 2)\) \((2 : -5 : 1)\) \((-2 : 7 : 1)\)

magma: [C![-2,3,1],C![-2,7,1],C![-1,1,1],C![0,-1,1],C![0,1,1],C![1,-4,2],C![1,-1,0],C![1,-1,2],C![1,0,0],C![2,-5,1]];
 

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

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((-1 : 1 : 1) + (0 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \(x (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-2xz^2 - z^3\) \(0.009642\) \(\infty\)
\((-1 : 1 : 1) + (2 : -5 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \((x - 2z) (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-2xz^2 - z^3\) \(0\) \(2\)

2-torsion field: 4.2.2092.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(2\)
Regulator: \( 0.009642 \)
Real period: \( 13.71617 \)
Tamagawa product: \( 12 \)
Torsion order:\( 2 \)
Leading coefficient: \( 0.396771 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(1\) \(6\) \(6\) \(( 1 - T )( 1 + 2 T + 2 T^{2} )\)
\(3\) \(1\) \(2\) \(2\) \(( 1 + T )( 1 + 2 T + 3 T^{2} )\)
\(523\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 - 4 T + 523 T^{2} )\)

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 \(\Q\).