Properties

Label 1497.b.13473.1
Conductor 1497
Discriminant 13473
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

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Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^3 + x + 1)y = -2x^5 + 3x^4 - x^2$ (homogenize, simplify)
$y^2 + (x^3 + xz^2 + z^3)y = -2x^5z + 3x^4z^2 - x^2z^4$ (dehomogenize, simplify)
$y^2 = x^6 - 8x^5 + 14x^4 + 2x^3 - 3x^2 + 2x + 1$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 0, -1, 0, 3, -2]), R([1, 1, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 0, -1, 0, 3, -2], R![1, 1, 0, 1]);
 
sage: X = HyperellipticCurve(R([1, 2, -3, 2, 14, -8, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(1497\) \(=\) \( 3 \cdot 499 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(13473\) \(=\) \( 3^{3} \cdot 499 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(92\) \(=\)  \( 2^{2} \cdot 23 \)
\( I_4 \)  \(=\) \(36025\) \(=\)  \( 5^{2} \cdot 11 \cdot 131 \)
\( I_6 \)  \(=\) \(3650051\) \(=\)  \( 3650051 \)
\( I_{10} \)  \(=\) \(-1724544\) \(=\)  \( - 2^{7} \cdot 3^{3} \cdot 499 \)
\( J_2 \)  \(=\) \(23\) \(=\)  \( 23 \)
\( J_4 \)  \(=\) \(-1479\) \(=\)  \( - 3 \cdot 17 \cdot 29 \)
\( J_6 \)  \(=\) \(-41077\) \(=\)  \( -41077 \)
\( J_8 \)  \(=\) \(-783053\) \(=\)  \( - 101 \cdot 7753 \)
\( J_{10} \)  \(=\) \(-13473\) \(=\)  \( - 3^{3} \cdot 499 \)
\( g_1 \)  \(=\) \(-6436343/13473\)
\( g_2 \)  \(=\) \(5998331/4491\)
\( g_3 \)  \(=\) \(21729733/13473\)

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
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 0 : 1)\) \((0 : -1 : 1)\) \((1 : 0 : 1)\) \((-1 : 0 : 2)\)
\((1 : -3 : 1)\) \((-1 : -3 : 2)\) \((-3 : -16 : 1)\) \((-3 : 45 : 1)\)

magma: [C![-3,-16,1],C![-3,45,1],C![-1,-3,2],C![-1,0,2],C![0,-1,1],C![0,0,1],C![1,-3,1],C![1,-1,0],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: \(\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((1 : -3 : 1) - (1 : -1 : 0)\) \(z (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(-3z^3\) \(0.002757\) \(\infty\)

2-torsion field: 6.2.95808.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(1\)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(1\)
Regulator: \( 0.002757 \)
Real period: \( 24.92045 \)
Tamagawa product: \( 3 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.206128 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(3\) \(1\) \(3\) \(3\) \(( 1 - T )( 1 + 2 T + 3 T^{2} )\)
\(499\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 10 T + 499 T^{2} )\)

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 \(\Q\).