Properties

Label 7927.b.7927.1
Conductor 7927
Discriminant -7927
Mordell-Weil group \(\Z \times \Z\)
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

$y^2 + x^3y = -2x^4 - x^3 + 3x^2 + x - 1$ (homogenize, simplify)
$y^2 + x^3y = -2x^4z^2 - x^3z^3 + 3x^2z^4 + xz^5 - z^6$ (dehomogenize, simplify)
$y^2 = x^6 - 8x^4 - 4x^3 + 12x^2 + 4x - 4$ (minimize, homogenize)

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

Invariants

Conductor: \( N \)  \(=\)  \(7927\) \(=\) \( 7927 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-7927\) \(=\) \( -7927 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(2592\) \(=\)  \( 2^{5} \cdot 3^{4} \)
\( I_4 \)  \(=\) \(38208\) \(=\)  \( 2^{6} \cdot 3 \cdot 199 \)
\( I_6 \)  \(=\) \(33408000\) \(=\)  \( 2^{10} \cdot 3^{2} \cdot 5^{3} \cdot 29 \)
\( I_{10} \)  \(=\) \(-32468992\) \(=\)  \( - 2^{12} \cdot 7927 \)
\( J_2 \)  \(=\) \(324\) \(=\)  \( 2^{2} \cdot 3^{4} \)
\( J_4 \)  \(=\) \(3976\) \(=\)  \( 2^{3} \cdot 7 \cdot 71 \)
\( J_6 \)  \(=\) \(56552\) \(=\)  \( 2^{3} \cdot 7069 \)
\( J_8 \)  \(=\) \(628568\) \(=\)  \( 2^{3} \cdot 78571 \)
\( J_{10} \)  \(=\) \(-7927\) \(=\)  \( -7927 \)
\( g_1 \)  \(=\) \(-3570467226624/7927\)
\( g_2 \)  \(=\) \(-135232602624/7927\)
\( g_3 \)  \(=\) \(-5936602752/7927\)

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

Known points
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((-1 : 0 : 1)\) \((1 : 0 : 1)\) \((-1 : 1 : 1)\) \((1 : -1 : 1)\)
\((1 : 0 : 2)\) \((1 : -1 : 2)\) \((-2 : 3 : 1)\) \((-2 : 5 : 1)\)

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

Number of rational Weierstrass points: \(0\)

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: \(\Z \times \Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : -1 : 0)\) \(z^2\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0.265853\) \(\infty\)
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 + xz - z^2\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2\) \(0.069502\) \(\infty\)

2-torsion field: 6.4.507328.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(2\)
Mordell-Weil rank: \(2\)
2-Selmer rank:\(2\)
Regulator: \( 0.018265 \)
Real period: \( 21.11808 \)
Tamagawa product: \( 1 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.385723 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(7927\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 91 T + 7927 T^{2} )\)

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 \(\Q\).