Properties

Label 5364.a.193104.1
Conductor 5364
Discriminant -193104
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

$y^2 + (x^3 + x)y = 2x^3 + 3x^2 + 2x + 1$ (homogenize, simplify)
$y^2 + (x^3 + xz^2)y = 2x^3z^3 + 3x^2z^4 + 2xz^5 + z^6$ (dehomogenize, simplify)
$y^2 = x^6 + 2x^4 + 8x^3 + 13x^2 + 8x + 4$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([1, 2, 3, 2]), R([0, 1, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![1, 2, 3, 2], R![0, 1, 0, 1]);
 
sage: X = HyperellipticCurve(R([4, 8, 13, 8, 2, 0, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(5364\) \(=\) \( 2^{2} \cdot 3^{2} \cdot 149 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-193104\) \(=\) \( - 2^{4} \cdot 3^{4} \cdot 149 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(-992\) \(=\)  \( - 2^{5} \cdot 31 \)
\( I_4 \)  \(=\) \(51712\) \(=\)  \( 2^{9} \cdot 101 \)
\( I_6 \)  \(=\) \(-10326272\) \(=\)  \( - 2^{8} \cdot 11 \cdot 19 \cdot 193 \)
\( I_{10} \)  \(=\) \(-790953984\) \(=\)  \( - 2^{16} \cdot 3^{4} \cdot 149 \)
\( J_2 \)  \(=\) \(-124\) \(=\)  \( - 2^{2} \cdot 31 \)
\( J_4 \)  \(=\) \(102\) \(=\)  \( 2 \cdot 3 \cdot 17 \)
\( J_6 \)  \(=\) \(-5040\) \(=\)  \( - 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \)
\( J_8 \)  \(=\) \(153639\) \(=\)  \( 3^{2} \cdot 43 \cdot 397 \)
\( J_{10} \)  \(=\) \(-193104\) \(=\)  \( - 2^{4} \cdot 3^{4} \cdot 149 \)
\( g_1 \)  \(=\) \(1832265664/12069\)
\( g_2 \)  \(=\) \(4051576/4023\)
\( g_3 \)  \(=\) \(538160/1341\)

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
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((-1 : 0 : 1)\) \((0 : -1 : 1)\) \((0 : 1 : 1)\) \((-1 : 2 : 1)\)
\((1 : 2 : 1)\) \((1 : -4 : 1)\) \((-11 : 106 : 5)\) \((-11 : 1500 : 5)\)

magma: [C![-11,106,5],C![-11,1500,5],C![-1,0,1],C![-1,2,1],C![0,-1,1],C![0,1,1],C![1,-4,1],C![1,-1,0],C![1,0,0],C![1,2,1]];
 

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
\((-1 : 0 : 1) - (1 : 0 : 0)\) \(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 - z^3\) \(0.003006\) \(\infty\)

2-torsion field: 6.0.343296.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(1\)
Regulator: \( 0.003006 \)
Real period: \( 12.93060 \)
Tamagawa product: \( 15 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.583063 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(2\) \(4\) \(3\) \(1 + T + 2 T^{2}\)
\(3\) \(2\) \(4\) \(5\) \(( 1 - T )( 1 + T )\)
\(149\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 - 15 T + 149 T^{2} )\)

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 \(\Q\).