Properties

Label 1472.a.94208.1
Conductor 1472
Discriminant -94208
Mordell-Weil group \(\Z/{2}\Z \times \Z/{2}\Z\)
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

Related objects

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Minimal equation

Minimal equation

Simplified equation

$y^2 = 4x^5 - 3x^4 - 4x^3 - x^2 + 7x - 3$ (homogenize, simplify)
$y^2 = 4x^5z - 3x^4z^2 - 4x^3z^3 - x^2z^4 + 7xz^5 - 3z^6$ (dehomogenize, simplify)
$y^2 = 4x^5 - 3x^4 - 4x^3 - x^2 + 7x - 3$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-3, 7, -1, -4, -3, 4]), R([]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-3, 7, -1, -4, -3, 4], R![]);
 
sage: X = HyperellipticCurve(R([-3, 7, -1, -4, -3, 4]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(1472\) \(=\) \( 2^{6} \cdot 23 \)
magma: Conductor(LSeries(C: ExcFactors:=[*<2,Valuation(1472,2),R![1]>*])); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-94208\) \(=\) \( - 2^{12} \cdot 23 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(18688\) \(=\)  \( 2^{8} \cdot 73 \)
\( I_4 \)  \(=\) \(308224\) \(=\)  \( 2^{10} \cdot 7 \cdot 43 \)
\( I_6 \)  \(=\) \(1560887296\) \(=\)  \( 2^{14} \cdot 47 \cdot 2027 \)
\( I_{10} \)  \(=\) \(-385875968\) \(=\)  \( - 2^{24} \cdot 23 \)
\( J_2 \)  \(=\) \(2336\) \(=\)  \( 2^{5} \cdot 73 \)
\( J_4 \)  \(=\) \(224160\) \(=\)  \( 2^{5} \cdot 3 \cdot 5 \cdot 467 \)
\( J_6 \)  \(=\) \(28881152\) \(=\)  \( 2^{8} \cdot 101 \cdot 1117 \)
\( J_8 \)  \(=\) \(4304666368\) \(=\)  \( 2^{8} \cdot 109 \cdot 154267 \)
\( J_{10} \)  \(=\) \(-94208\) \(=\)  \( - 2^{12} \cdot 23 \)
\( g_1 \)  \(=\) \(-16982602489856/23\)
\( g_2 \)  \(=\) \(-697616405760/23\)
\( g_3 \)  \(=\) \(-38476914752/23\)

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: \((1 : 0 : 0),\, (1 : 0 : 1),\, (3 : 0 : 4)\)

magma: [C![1,0,0],C![1,0,1],C![3,0,4]];
 

Number of rational Weierstrass points: \(3\)

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

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((1 : 0 : 1) - (1 : 0 : 0)\) \(x - z\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(2\)
\((1 : 0 : 1) + (3 : 0 : 4) - 2 \cdot(1 : 0 : 0)\) \((x - z) (4x - 3z)\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(2\)

2-torsion field: 3.1.23.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(2\)
Regulator: \( 1 \)
Real period: \( 3.929409 \)
Tamagawa product: \( 2 \)
Torsion order:\( 4 \)
Leading coefficient: \( 0.491176 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(6\) \(12\) \(2\) \(1\)
\(23\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 - 8 T + 23 T^{2} )\)

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 \(\Q\).