Properties

Label 41411.a.41411.1
Conductor 41411
Discriminant 41411
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

$y^2 + (x^3 + 1)y = -3x^4 + 7x^3 - 4x^2$ (homogenize, simplify)
$y^2 + (x^3 + z^3)y = -3x^4z^2 + 7x^3z^3 - 4x^2z^4$ (dehomogenize, simplify)
$y^2 = x^6 - 12x^4 + 30x^3 - 16x^2 + 1$ (minimize, homogenize)

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 0, -4, 7, -3], R![1, 0, 0, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 0, -4, 7, -3]), R([1, 0, 0, 1]));
 
magma: X,pi:= SimplifiedModel(C);
 
sage: X = HyperellipticCurve(R([1, 0, -16, 30, -12, 0, 1]))
 

Invariants

Conductor: \( N \)  =  \(41411\) = \( 41411 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(41411\) = \( 41411 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(2088\) =  \( 2^{3} \cdot 3^{2} \cdot 29 \)
\( I_4 \)  = \(64356\) =  \( 2^{2} \cdot 3 \cdot 31 \cdot 173 \)
\( I_6 \)  = \(21315816\) =  \( 2^{3} \cdot 3^{2} \cdot 47 \cdot 6299 \)
\( I_{10} \)  = \(169619456\) =  \( 2^{12} \cdot 41411 \)
\( J_2 \)  = \(261\) =  \( 3^{2} \cdot 29 \)
\( J_4 \)  = \(2168\) =  \( 2^{3} \cdot 271 \)
\( J_6 \)  = \(52752\) =  \( 2^{4} \cdot 3 \cdot 7 \cdot 157 \)
\( J_8 \)  = \(2267012\) =  \( 2^{2} \cdot 11 \cdot 67 \cdot 769 \)
\( J_{10} \)  = \(41411\) =  \( 41411 \)
\( g_1 \)  = \(1211162837301/41411\)
\( g_2 \)  = \(38546131608/41411\)
\( g_3 \)  = \(3593518992/41411\)

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
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 0 : 1)\) \((0 : -1 : 1)\) \((1 : 0 : 1)\) \((1 : -2 : 1)\)
\((2 : -1 : 1)\) \((1 : -4 : 2)\) \((4 : 0 : 3)\) \((1 : -5 : 2)\) \((2 : -8 : 1)\) \((2 : -8 : 3)\)
\((2 : -27 : 3)\) \((4 : -91 : 3)\) \((3 : -252 : 10)\) \((3 : -775 : 10)\) \((-23 : -525850 : 185)\) \((-23 : -5793608 : 185)\)

magma: [C![-23,-5793608,185],C![-23,-525850,185],C![0,-1,1],C![0,0,1],C![1,-5,2],C![1,-4,2],C![1,-2,1],C![1,-1,0],C![1,0,0],C![1,0,1],C![2,-27,3],C![2,-8,1],C![2,-8,3],C![2,-1,1],C![3,-775,10],C![3,-252,10],C![4,-91,3],C![4,0,3]];
 

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
\((0 : -1 : 1) - (1 : 0 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 - z^3\) \(0.488627\) \(\infty\)
\((1 : -5 : 2) + (1 : -2 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \((x - z) (2x - z)\) \(=\) \(0,\) \(4y\) \(=\) \(-11xz^2 + 3z^3\) \(0.587592\) \(\infty\)
\((1 : -5 : 2) - (1 : 0 : 0)\) \(z (2x - z)\) \(=\) \(0,\) \(2y\) \(=\) \(-2x^3 - z^3\) \(0.109268\) \(\infty\)

2-torsion field: 6.2.2650304.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(3\)   (upper bound)
Mordell-Weil rank: \(3\)
2-Selmer rank:\(3\)
Regulator: \( 0.029860 \)
Real period: \( 19.86418 \)
Tamagawa product: \( 1 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.593147 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(41411\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 - 97 T + 41411 T^{2} )\)

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 \(\Q\).