Properties

Label 54983.a.54983.1
Conductor 54983
Discriminant -54983
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

Learn more about

Show commands for: Magma / SageMath

Minimal equation

Minimal equation

Simplified equation

$y^2 + y = x^6 - x^5 - 3x^4 + x^3 + 3x^2 + x$ (homogenize, simplify)
$y^2 + z^3y = x^6 - x^5z - 3x^4z^2 + x^3z^3 + 3x^2z^4 + xz^5$ (dehomogenize, simplify)
$y^2 = 4x^6 - 4x^5 - 12x^4 + 4x^3 + 12x^2 + 4x + 1$ (minimize, homogenize)

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 1, 3, 1, -3, -1, 1], R![1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 1, 3, 1, -3, -1, 1]), R([1]));
 
magma: X,pi:= SimplifiedModel(C);
 
sage: X = HyperellipticCurve(R([1, 4, 12, 4, -12, -4, 4]))
 

Invariants

Conductor: \( N \)  =  \(54983\) = \( 54983 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(-54983\) = \( - 54983 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(800\) =  \( 2^{5} \cdot 5^{2} \)
\( I_4 \)  = \(337216\) =  \( 2^{6} \cdot 11 \cdot 479 \)
\( I_6 \)  = \(41495552\) =  \( 2^{10} \cdot 7^{2} \cdot 827 \)
\( I_{10} \)  = \(-225210368\) =  \( - 2^{12} \cdot 54983 \)
\( J_2 \)  = \(100\) =  \( 2^{2} \cdot 5^{2} \)
\( J_4 \)  = \(-3096\) =  \( - 2^{3} \cdot 3^{2} \cdot 43 \)
\( J_6 \)  = \(27848\) =  \( 2^{3} \cdot 59^{2} \)
\( J_8 \)  = \(-1700104\) =  \( - 2^{3} \cdot 7^{2} \cdot 4337 \)
\( J_{10} \)  = \(-54983\) =  \( - 54983 \)
\( g_1 \)  = \(-10000000000/54983\)
\( g_2 \)  = \(3096000000/54983\)
\( g_3 \)  = \(-278480000/54983\)

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
\((1 : -1 : 0)\) \((1 : 1 : 0)\) \((0 : 0 : 1)\) \((-1 : 0 : 1)\) \((0 : -1 : 1)\) \((-1 : -1 : 1)\)
\((1 : 1 : 1)\) \((1 : -2 : 1)\) \((2 : 2 : 1)\) \((-1 : -2 : 3)\) \((2 : -3 : 1)\) \((-4 : -4 : 5)\)
\((-1 : -25 : 3)\) \((-4 : -121 : 5)\) \((5 : 3721 : 24)\) \((5 : -17545 : 24)\) \((54 : 129735 : 11)\) \((54 : -131066 : 11)\)

magma: [C![-4,-121,5],C![-4,-4,5],C![-1,-25,3],C![-1,-2,3],C![-1,-1,1],C![-1,0,1],C![0,-1,1],C![0,0,1],C![1,-2,1],C![1,-1,0],C![1,1,0],C![1,1,1],C![2,-3,1],C![2,2,1],C![5,-17545,24],C![5,3721,24],C![54,-131066,11],C![54,129735,11]];
 

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
\((1 : -2 : 1) - (1 : -1 : 0)\) \(z (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(x^3 - 3z^3\) \(0.628570\) \(\infty\)
\((0 : -1 : 1) - (1 : -1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(x^3 - z^3\) \(0.432242\) \(\infty\)
\(D_0 - (1 : -1 : 0) - (1 : 1 : 0)\) \(x^2 - xz - z^2\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.145844\) \(\infty\)

2-torsion field: 6.0.3518912.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(3\)   (upper bound)
Mordell-Weil rank: \(3\)
2-Selmer rank:\(3\)
Regulator: \( 0.032929 \)
Real period: \( 21.51843 \)
Tamagawa product: \( 1 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.708582 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(54983\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 - 176 T + 54983 T^{2} )\)

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 \(\Q\).