Properties

Label 4128.b.594432.1
Conductor 4128
Discriminant -594432
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

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Minimal equation

Minimal equation

Simplified equation

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

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

Invariants

Conductor: \( N \)  \(=\)  \(4128\) \(=\) \( 2^{5} \cdot 3 \cdot 43 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-594432\) \(=\) \( - 2^{9} \cdot 3^{3} \cdot 43 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(-864\) \(=\)  \( - 2^{5} \cdot 3^{3} \)
\( I_4 \)  \(=\) \(105024\) \(=\)  \( 2^{6} \cdot 3 \cdot 547 \)
\( I_6 \)  \(=\) \(-25155072\) \(=\)  \( - 2^{9} \cdot 3^{2} \cdot 53 \cdot 103 \)
\( I_{10} \)  \(=\) \(-2434793472\) \(=\)  \( - 2^{21} \cdot 3^{3} \cdot 43 \)
\( J_2 \)  \(=\) \(-108\) \(=\)  \( - 2^{2} \cdot 3^{3} \)
\( J_4 \)  \(=\) \(-608\) \(=\)  \( - 2^{5} \cdot 19 \)
\( J_6 \)  \(=\) \(7936\) \(=\)  \( 2^{8} \cdot 31 \)
\( J_8 \)  \(=\) \(-306688\) \(=\)  \( - 2^{9} \cdot 599 \)
\( J_{10} \)  \(=\) \(-594432\) \(=\)  \( - 2^{9} \cdot 3^{3} \cdot 43 \)
\( g_1 \)  \(=\) \(1062882/43\)
\( g_2 \)  \(=\) \(-55404/43\)
\( g_3 \)  \(=\) \(-6696/43\)

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

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

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
\((-1 : -7 : 2) + (1 : 1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) \((x - z) (2x + z)\) \(=\) \(0,\) \(4y\) \(=\) \(5xz^2 - z^3\) \(0.001730\) \(\infty\)

2-torsion field: 6.0.264192.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(1\)
Regulator: \( 0.001730 \)
Real period: \( 13.79653 \)
Tamagawa product: \( 24 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.572871 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(5\) \(9\) \(8\) \(1\)
\(3\) \(1\) \(3\) \(3\) \(( 1 - T )( 1 + 3 T + 3 T^{2} )\)
\(43\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 4 T + 43 T^{2} )\)

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 \(\Q\).