Properties

Label 100154.a.200308.1
Conductor 100154
Discriminant 200308
Mordell-Weil group \(\Z \times \Z\)
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

$y^2 + (x + 1)y = x^6 + x^5 + 2x^4 + 3x^3 + x^2$ (homogenize, simplify)
$y^2 + (xz^2 + z^3)y = x^6 + x^5z + 2x^4z^2 + 3x^3z^3 + x^2z^4$ (dehomogenize, simplify)
$y^2 = 4x^6 + 4x^5 + 8x^4 + 12x^3 + 5x^2 + 2x + 1$ (minimize, homogenize)

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 0, 1, 3, 2, 1, 1], R![1, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 0, 1, 3, 2, 1, 1]), R([1, 1]));
 
magma: X,pi:= SimplifiedModel(C);
 
sage: X = HyperellipticCurve(R([1, 2, 5, 12, 8, 4, 4]))
 

Invariants

Conductor: \( N \)  =  \(100154\) = \( 2 \cdot 50077 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(200308\) = \( 2^{2} \cdot 50077 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(-416\) =  \( - 2^{5} \cdot 13 \)
\( I_4 \)  = \(81280\) =  \( 2^{7} \cdot 5 \cdot 127 \)
\( I_6 \)  = \(-8619328\) =  \( - 2^{6} \cdot 134677 \)
\( I_{10} \)  = \(820461568\) =  \( 2^{14} \cdot 50077 \)
\( J_2 \)  = \(-52\) =  \( - 2^{2} \cdot 13 \)
\( J_4 \)  = \(-734\) =  \( - 2 \cdot 367 \)
\( J_6 \)  = \(2409\) =  \( 3 \cdot 11 \cdot 73 \)
\( J_8 \)  = \(-166006\) =  \( - 2 \cdot 83003 \)
\( J_{10} \)  = \(200308\) =  \( 2^{2} \cdot 50077 \)
\( g_1 \)  = \(-95051008/50077\)
\( g_2 \)  = \(25801568/50077\)
\( g_3 \)  = \(1628484/50077\)

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: \((1 : -1 : 0),\, (1 : 1 : 0),\, (0 : 0 : 1),\, (-1 : 0 : 1),\, (0 : -1 : 1),\, (1 : 2 : 1),\, (1 : -4 : 1)\)

magma: [C![-1,0,1],C![0,-1,1],C![0,0,1],C![1,-4,1],C![1,-1,0],C![1,1,0],C![1,2,1]];
 

Number of rational Weierstrass points: \(1\)

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: \(\Z \times \Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((0 : -1 : 1) - (1 : 1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 - z^3\) \(0.497498\) \(\infty\)
\((-1 : 0 : 1) + (0 : -1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) \(x (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2 - z^3\) \(0.103529\) \(\infty\)

2-torsion field: 5.1.801232.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(2\)
Mordell-Weil rank: \(2\)
2-Selmer rank:\(2\)
Regulator: \( 0.051351 \)
Real period: \( 13.92194 \)
Tamagawa product: \( 2 \)
Torsion order:\( 1 \)
Leading coefficient: \( 1.429825 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(2\) \(2\) \(1\) \(2\) \(( 1 + T )( 1 + 2 T^{2} )\)
\(50077\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 - 120 T + 50077 T^{2} )\)

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 \(\Q\).