Properties

Label 7529.a.7529.1
Conductor 7529
Discriminant -7529
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

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Minimal equation

Minimal equation

Simplified equation

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

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([1, 2, 1]), R([0, 1, 1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![1, 2, 1], R![0, 1, 1, 1]);
 
sage: X = HyperellipticCurve(R([4, 8, 5, 2, 3, 2, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(7529\) \(=\) \( 7529 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-7529\) \(=\) \( -7529 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(-536\) \(=\)  \( - 2^{3} \cdot 67 \)
\( I_4 \)  \(=\) \(16804\) \(=\)  \( 2^{2} \cdot 4201 \)
\( I_6 \)  \(=\) \(-2369624\) \(=\)  \( - 2^{3} \cdot 271 \cdot 1093 \)
\( I_{10} \)  \(=\) \(-30838784\) \(=\)  \( - 2^{12} \cdot 7529 \)
\( J_2 \)  \(=\) \(-67\) \(=\)  \( -67 \)
\( J_4 \)  \(=\) \(12\) \(=\)  \( 2^{2} \cdot 3 \)
\( J_6 \)  \(=\) \(160\) \(=\)  \( 2^{5} \cdot 5 \)
\( J_8 \)  \(=\) \(-2716\) \(=\)  \( - 2^{2} \cdot 7 \cdot 97 \)
\( J_{10} \)  \(=\) \(-7529\) \(=\)  \( -7529 \)
\( g_1 \)  \(=\) \(1350125107/7529\)
\( g_2 \)  \(=\) \(3609156/7529\)
\( g_3 \)  \(=\) \(-718240/7529\)

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
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((-1 : 0 : 1)\) \((0 : -1 : 1)\) \((0 : 1 : 1)\) \((-1 : 1 : 1)\)
\((1 : 1 : 1)\) \((1 : -4 : 1)\) \((1 : 9 : 2)\) \((1 : -16 : 2)\)

magma: [C![-1,0,1],C![-1,1,1],C![0,-1,1],C![0,1,1],C![1,-16,2],C![1,-4,1],C![1,-1,0],C![1,0,0],C![1,1,1],C![1,9,2]];
 

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
\((0 : -1 : 1) - (1 : 0 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 - z^3\) \(0.460889\) \(\infty\)
\((-1 : 1 : 1) - (1 : 0 : 0)\) \(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-x^3\) \(0.039904\) \(\infty\)

2-torsion field: 6.0.481856.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(2\)
Mordell-Weil rank: \(2\)
2-Selmer rank:\(2\)
Regulator: \( 0.018194 \)
Real period: \( 19.21437 \)
Tamagawa product: \( 1 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.349604 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(7529\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 15 T + 7529 T^{2} )\)

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 \(\Q\).