Properties

Label 57065.a.285325.1
Conductor 57065
Discriminant -285325
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

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Minimal equation

Minimal equation

Simplified equation

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

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 0, 1, -4, 1], R![1, 1, 0, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 0, 1, -4, 1]), R([1, 1, 0, 1]));
 
magma: X,pi:= SimplifiedModel(C);
 
sage: X = HyperellipticCurve(R([1, 2, 5, -14, 6, 0, 1]))
 

Invariants

Conductor: \( N \)  =  \(57065\) = \( 5 \cdot 101 \cdot 113 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(-285325\) = \( - 5^{2} \cdot 101 \cdot 113 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(456\) =  \( 2^{3} \cdot 3 \cdot 19 \)
\( I_4 \)  = \(114468\) =  \( 2^{2} \cdot 3 \cdot 9539 \)
\( I_6 \)  = \(11869608\) =  \( 2^{3} \cdot 3 \cdot 494567 \)
\( I_{10} \)  = \(-1168691200\) =  \( - 2^{12} \cdot 5^{2} \cdot 101 \cdot 113 \)
\( J_2 \)  = \(57\) =  \( 3 \cdot 19 \)
\( J_4 \)  = \(-1057\) =  \( - 7 \cdot 151 \)
\( J_6 \)  = \(-1299\) =  \( - 3 \cdot 433 \)
\( J_8 \)  = \(-297823\) =  \( - 17 \cdot 17519 \)
\( J_{10} \)  = \(-285325\) =  \( - 5^{2} \cdot 101 \cdot 113 \)
\( g_1 \)  = \(-601692057/285325\)
\( g_2 \)  = \(195749001/285325\)
\( g_3 \)  = \(4220451/285325\)

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
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 0 : 1)\) \((0 : -1 : 1)\) \((1 : -1 : 1)\) \((-1 : -2 : 1)\)
\((1 : -2 : 1)\) \((1 : -1 : 2)\) \((-1 : 3 : 1)\) \((-2 : -4 : 1)\) \((2 : -9 : 3)\) \((3 : -11 : 2)\)
\((1 : -12 : 2)\) \((-2 : 13 : 1)\) \((-2 : 36 : 3)\) \((3 : -36 : 2)\) \((-2 : -37 : 3)\) \((2 : -44 : 3)\)

magma: [C![-2,-37,3],C![-2,-4,1],C![-2,13,1],C![-2,36,3],C![-1,-2,1],C![-1,3,1],C![0,-1,1],C![0,0,1],C![1,-12,2],C![1,-2,1],C![1,-1,0],C![1,-1,1],C![1,-1,2],C![1,0,0],C![2,-44,3],C![2,-9,3],C![3,-36,2],C![3,-11,2]];
 

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
\((-1 : -2 : 1) - (1 : -1 : 0)\) \(z (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-2z^3\) \(0.400353\) \(\infty\)
\((0 : 0 : 1) - (1 : -1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0.498323\) \(\infty\)
\((-1 : -2 : 1) + (1 : -2 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \((x - z) (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-2z^3\) \(0.097232\) \(\infty\)

2-torsion field: 6.0.730432.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(3\)   (upper bound)
Mordell-Weil rank: \(3\)
2-Selmer rank:\(3\)
Regulator: \( 0.016342 \)
Real period: \( 17.23534 \)
Tamagawa product: \( 2 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.563352 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(5\) \(2\) \(1\) \(2\) \(( 1 + T )( 1 + 2 T + 5 T^{2} )\)
\(101\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 8 T + 101 T^{2} )\)
\(113\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 2 T + 113 T^{2} )\)

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 \(\Q\).