Properties

Label 3528.b.338688.1
Conductor 3528
Discriminant -338688
Mordell-Weil group \(\Z\)
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

Learn more about

Show commands for: SageMath / Magma

Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^3 + x^2 + x + 1)y = 2x^4 - x^3 - x$ (homogenize, simplify)
$y^2 + (x^3 + x^2z + xz^2 + z^3)y = 2x^4z^2 - x^3z^3 - xz^5$ (dehomogenize, simplify)
$y^2 = x^6 + 2x^5 + 11x^4 + 3x^2 - 2x + 1$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -1, 0, -1, 2]), R([1, 1, 1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -1, 0, -1, 2], R![1, 1, 1, 1]);
 
sage: X = HyperellipticCurve(R([1, -2, 3, 0, 11, 2, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(3528\) \(=\) \( 2^{3} \cdot 3^{2} \cdot 7^{2} \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-338688\) \(=\) \( - 2^{8} \cdot 3^{3} \cdot 7^{2} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(-928\) \(=\)  \( - 2^{5} \cdot 29 \)
\( I_4 \)  \(=\) \(71680\) \(=\)  \( 2^{11} \cdot 5 \cdot 7 \)
\( I_6 \)  \(=\) \(-16457728\) \(=\)  \( - 2^{13} \cdot 7^{2} \cdot 41 \)
\( I_{10} \)  \(=\) \(-1387266048\) \(=\)  \( - 2^{20} \cdot 3^{3} \cdot 7^{2} \)
\( J_2 \)  \(=\) \(-116\) \(=\)  \( - 2^{2} \cdot 29 \)
\( J_4 \)  \(=\) \(-186\) \(=\)  \( - 2 \cdot 3 \cdot 31 \)
\( J_6 \)  \(=\) \(900\) \(=\)  \( 2^{2} \cdot 3^{2} \cdot 5^{2} \)
\( J_8 \)  \(=\) \(-34749\) \(=\)  \( - 3^{5} \cdot 11 \cdot 13 \)
\( J_{10} \)  \(=\) \(-338688\) \(=\)  \( - 2^{8} \cdot 3^{3} \cdot 7^{2} \)
\( g_1 \)  \(=\) \(82044596/1323\)
\( g_2 \)  \(=\) \(-756059/882\)
\( g_3 \)  \(=\) \(-21025/588\)

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
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 0 : 1)\) \((0 : -1 : 1)\) \((1 : 0 : 1)\) \((-1 : -2 : 1)\)
\((-1 : 2 : 1)\) \((1 : -4 : 1)\) \((-3 : 100 : 5)\) \((-3 : -168 : 5)\)

magma: [C![-3,-168,5],C![-3,100,5],C![-1,-2,1],C![-1,2,1],C![0,-1,1],C![0,0,1],C![1,-4,1],C![1,-1,0],C![1,0,0],C![1,0,1]];
 

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: \(\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((0 : -1 : 1) + (1 : 0 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \((-x + z) x\) \(=\) \(0,\) \(y\) \(=\) \(xz^2 - z^3\) \(0.002338\) \(\infty\)

2-torsion field: 6.0.21168.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(1\)
Regulator: \( 0.002338 \)
Real period: \( 13.49978 \)
Tamagawa product: \( 12 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.378881 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(3\) \(8\) \(4\) \(1 + 2 T + 2 T^{2}\)
\(3\) \(2\) \(3\) \(3\) \(1 + T + T^{2}\)
\(7\) \(2\) \(2\) \(1\) \(1 + 3 T + 7 T^{2}\)

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 \(\Q\).