Properties

Label 5026.a.35182.1
Conductor 5026
Discriminant 35182
Mordell-Weil group \(\Z \times \Z \times \Z/{2}\Z\)
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

$y^2 + (x^3 + 1)y = 4x^5 + 22x^4 + 46x^3 + 28x^2 + 5x$ (homogenize, simplify)
$y^2 + (x^3 + z^3)y = 4x^5z + 22x^4z^2 + 46x^3z^3 + 28x^2z^4 + 5xz^5$ (dehomogenize, simplify)
$y^2 = x^6 + 16x^5 + 88x^4 + 186x^3 + 112x^2 + 20x + 1$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 5, 28, 46, 22, 4]), R([1, 0, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 5, 28, 46, 22, 4], R![1, 0, 0, 1]);
 
sage: X = HyperellipticCurve(R([1, 20, 112, 186, 88, 16, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(5026\) \(=\) \( 2 \cdot 7 \cdot 359 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(35182\) \(=\) \( 2 \cdot 7^{2} \cdot 359 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(62440\) \(=\)  \( 2^{3} \cdot 5 \cdot 7 \cdot 223 \)
\( I_4 \)  \(=\) \(1113316\) \(=\)  \( 2^{2} \cdot 278329 \)
\( I_6 \)  \(=\) \(22822085992\) \(=\)  \( 2^{3} \cdot 2852760749 \)
\( I_{10} \)  \(=\) \(144105472\) \(=\)  \( 2^{13} \cdot 7^{2} \cdot 359 \)
\( J_2 \)  \(=\) \(7805\) \(=\)  \( 5 \cdot 7 \cdot 223 \)
\( J_4 \)  \(=\) \(2526654\) \(=\)  \( 2 \cdot 3 \cdot 13 \cdot 29 \cdot 1117 \)
\( J_6 \)  \(=\) \(1086135208\) \(=\)  \( 2^{3} \cdot 113 \cdot 491 \cdot 2447 \)
\( J_8 \)  \(=\) \(523326215681\) \(=\)  \( 41 \cdot 59 \cdot 2153 \cdot 100483 \)
\( J_{10} \)  \(=\) \(35182\) \(=\)  \( 2 \cdot 7^{2} \cdot 359 \)
\( g_1 \)  \(=\) \(591110204777028125/718\)
\( g_2 \)  \(=\) \(12258530733232875/359\)
\( g_3 \)  \(=\) \(675155221982900/359\)

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
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 0 : 1)\) \((0 : -1 : 1)\) \((-1 : 0 : 2)\) \((-1 : -7 : 2)\)
\((-4 : 28 : 1)\) \((-4 : 35 : 1)\) \((-6 : 105 : 1)\) \((-6 : 110 : 1)\)

magma: [C![-6,105,1],C![-6,110,1],C![-4,28,1],C![-4,35,1],C![-1,-7,2],C![-1,0,2],C![0,-1,1],C![0,0,1],C![1,-1,0],C![1,0,0]];
 

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

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 + 5xz - 3z^2\) \(=\) \(0,\) \(y\) \(=\) \(-15xz^2 + 3z^3\) \(0.428002\) \(\infty\)
\((-4 : 35 : 1) + (0 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \(x (x + 4z)\) \(=\) \(0,\) \(y\) \(=\) \(-9xz^2 - z^3\) \(0.075443\) \(\infty\)
\(D_0 - (1 : -1 : 0) - (1 : 0 : 0)\) \(x^2 + 6xz + z^2\) \(=\) \(0,\) \(2y\) \(=\) \(-35xz^2 - 7z^3\) \(0\) \(2\)

2-torsion field: 6.6.263948288.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(2\)
Mordell-Weil rank: \(2\)
2-Selmer rank:\(3\)
Regulator: \( 0.031070 \)
Real period: \( 18.25299 \)
Tamagawa product: \( 2 \)
Torsion order:\( 2 \)
Leading coefficient: \( 0.283562 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + T + 2 T^{2} )\)
\(7\) \(1\) \(2\) \(2\) \(( 1 + T )( 1 + 3 T + 7 T^{2} )\)
\(359\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 - 12 T + 359 T^{2} )\)

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 \(\Q\).