Properties

Label 59967.a.539703.1
Conductor 59967
Discriminant -539703
Mordell-Weil group \(\Z \times \Z \times \Z\)
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

$y^2 + (x^3 + x^2 + 1)y = x^4 + 2x^3 - x^2 - 2x$ (homogenize, simplify)
$y^2 + (x^3 + x^2z + z^3)y = x^4z^2 + 2x^3z^3 - x^2z^4 - 2xz^5$ (dehomogenize, simplify)
$y^2 = x^6 + 2x^5 + 5x^4 + 10x^3 - 2x^2 - 8x + 1$ (minimize, homogenize)

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -2, -1, 2, 1], R![1, 0, 1, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -2, -1, 2, 1]), R([1, 0, 1, 1]));
 
magma: X,pi:= SimplifiedModel(C);
 
sage: X = HyperellipticCurve(R([1, -8, -2, 10, 5, 2, 1]))
 

Invariants

Conductor: \( N \)  =  \(59967\) = \( 3^{3} \cdot 2221 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(-539703\) = \( - 3^{5} \cdot 2221 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(-120\) =  \( - 2^{3} \cdot 3 \cdot 5 \)
\( I_4 \)  = \(46116\) =  \( 2^{2} \cdot 3^{3} \cdot 7 \cdot 61 \)
\( I_6 \)  = \(-16924248\) =  \( - 2^{3} \cdot 3^{3} \cdot 11 \cdot 17 \cdot 419 \)
\( I_{10} \)  = \(-2210623488\) =  \( - 2^{12} \cdot 3^{5} \cdot 2221 \)
\( J_2 \)  = \(-15\) =  \( - 3 \cdot 5 \)
\( J_4 \)  = \(-471\) =  \( - 3 \cdot 157 \)
\( J_6 \)  = \(27373\) =  \( 31 \cdot 883 \)
\( J_8 \)  = \(-158109\) =  \( - 3 \cdot 7 \cdot 7529 \)
\( J_{10} \)  = \(-539703\) =  \( - 3^{5} \cdot 2221 \)
\( g_1 \)  = \(3125/2221\)
\( g_2 \)  = \(-19625/6663\)
\( g_3 \)  = \(-684325/59967\)

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
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 0 : 1)\) \((-1 : 0 : 1)\) \((0 : -1 : 1)\) \((1 : 0 : 1)\)
\((-1 : -1 : 1)\) \((-2 : 0 : 1)\) \((1 : -3 : 1)\) \((-2 : 3 : 1)\) \((-1 : 3 : 2)\) \((3 : 3 : 1)\)
\((3 : 7 : 2)\) \((-2 : 9 : 3)\) \((-1 : -12 : 2)\) \((-7 : -15 : 2)\) \((3 : -40 : 1)\) \((-2 : -40 : 3)\)
\((3 : -60 : 2)\) \((-7 : 252 : 2)\)

magma: [C![-7,-15,2],C![-7,252,2],C![-2,-40,3],C![-2,0,1],C![-2,3,1],C![-2,9,3],C![-1,-12,2],C![-1,-1,1],C![-1,0,1],C![-1,3,2],C![0,-1,1],C![0,0,1],C![1,-3,1],C![1,-1,0],C![1,0,0],C![1,0,1],C![3,-60,2],C![3,-40,1],C![3,3,1],C![3,7,2]];
 

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: \(\Z \times \Z \times \Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((-2 : 0 : 1) - (1 : -1 : 0)\) \(z (x + 2z)\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0.398690\) \(\infty\)
\((-1 : -1 : 1) + (1 : -3 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \((x - z) (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2 - 2z^3\) \(0.228205\) \(\infty\)
\((-2 : 0 : 1) + (0 : 0 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \(x (x + 2z)\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0.192414\) \(\infty\)

2-torsion field: 6.4.3837888.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(3\)   (upper bound)
Mordell-Weil rank: \(3\)
2-Selmer rank:\(3\)
Regulator: \( 0.013598 \)
Real period: \( 14.89874 \)
Tamagawa product: \( 3 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.607807 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(3\) \(5\) \(3\) \(3\) \(1 + 3 T + 3 T^{2}\)
\(2221\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 - 76 T + 2221 T^{2} )\)

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 \(\Q\).