Properties

Label 4925.b.4925.1
Conductor 4925
Discriminant 4925
Mordell-Weil group \(\Z \times \Z\)
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

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

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

Invariants

Conductor: \( N \)  =  \(4925\) = \( 5^{2} \cdot 197 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(4925\) = \( 5^{2} \cdot 197 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(-864\) =  \( - 2^{5} \cdot 3^{3} \)
\( I_4 \)  = \(18240\) =  \( 2^{6} \cdot 3 \cdot 5 \cdot 19 \)
\( I_6 \)  = \(-5529600\) =  \( - 2^{13} \cdot 3^{3} \cdot 5^{2} \)
\( I_{10} \)  = \(20172800\) =  \( 2^{12} \cdot 5^{2} \cdot 197 \)
\( J_2 \)  = \(-108\) =  \( - 2^{2} \cdot 3^{3} \)
\( J_4 \)  = \(296\) =  \( 2^{3} \cdot 37 \)
\( J_6 \)  = \(984\) =  \( 2^{3} \cdot 3 \cdot 41 \)
\( J_8 \)  = \(-48472\) =  \( - 2^{3} \cdot 73 \cdot 83 \)
\( J_{10} \)  = \(4925\) =  \( 5^{2} \cdot 197 \)
\( g_1 \)  = \(-14693280768/4925\)
\( g_2 \)  = \(-372874752/4925\)
\( g_3 \)  = \(11477376/4925\)

magma: IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);
 
sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]
 

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
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((-1 : 0 : 1)\) \((0 : -1 : 1)\) \((0 : 1 : 1)\) \((1 : 0 : 1)\)
\((-1 : 1 : 1)\) \((1 : -1 : 1)\) \((-3 : 7 : 2)\) \((-3 : 20 : 2)\) \((-11 : 260 : 6)\) \((-11 : 1071 : 6)\)

magma: [C![-11,260,6],C![-11,1071,6],C![-3,7,2],C![-3,20,2],C![-1,0,1],C![-1,1,1],C![0,-1,1],C![0,1,1],C![1,-1,0],C![1,-1,1],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: \(\Z \times \Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((-1 : 0 : 1) - (1 : 0 : 0)\) \(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 - z^3\) \(0.179718\) \(\infty\)
\(D_0 - 2 \cdot(1 : -1 : 0)\) \(z^2\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0.072321\) \(\infty\)

2-torsion field: 6.2.315200.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(2\)
Mordell-Weil rank: \(2\)
2-Selmer rank:\(2\)
Regulator: \( 0.012038 \)
Real period: \( 23.18140 \)
Tamagawa product: \( 1 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.279065 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(5\) \(2\) \(2\) \(1\) \(1 + 3 T + 5 T^{2}\)
\(197\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 - 18 T + 197 T^{2} )\)

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 \(\Q\).