Properties

Label 100017.a.300051.1
Conductor 100017
Discriminant 300051
Mordell-Weil group \(\Z/{3}\Z\)
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

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

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-1, 1, 0, 1], R![1, 1, 0, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-1, 1, 0, 1]), R([1, 1, 0, 1]));
 
magma: X,pi:= SimplifiedModel(C);
 
sage: X = HyperellipticCurve(R([-3, 6, 1, 6, 2, 0, 1]))
 

Invariants

Conductor: \( N \)  =  \(100017\) = \( 3^{2} \cdot 11113 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(300051\) = \( 3^{3} \cdot 11113 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(904\) =  \( 2^{3} \cdot 113 \)
\( I_4 \)  = \(-5084\) =  \( - 2^{2} \cdot 31 \cdot 41 \)
\( I_6 \)  = \(7140520\) =  \( 2^{3} \cdot 5 \cdot 178513 \)
\( I_{10} \)  = \(1229008896\) =  \( 2^{12} \cdot 3^{3} \cdot 11113 \)
\( J_2 \)  = \(113\) =  \( 113 \)
\( J_4 \)  = \(585\) =  \( 3^{2} \cdot 5 \cdot 13 \)
\( J_6 \)  = \(-10719\) =  \( - 3^{3} \cdot 397 \)
\( J_8 \)  = \(-388368\) =  \( - 2^{4} \cdot 3^{3} \cdot 29 \cdot 31 \)
\( J_{10} \)  = \(300051\) =  \( 3^{3} \cdot 11113 \)
\( g_1 \)  = \(18424351793/300051\)
\( g_2 \)  = \(93788305/33339\)
\( g_3 \)  = \(-5069293/11113\)

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

All points: \((1 : 0 : 0),\, (1 : -1 : 0)\)

magma: [C![1,-1,0],C![1,0,0]];
 

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: \(\Z/{3}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : -1 : 0)\) \(z^2\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(3\)

2-torsion field: 6.2.19203264.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(0\)
Regulator: \( 1 \)
Real period: \( 8.897841 \)
Tamagawa product: \( 3 \)
Torsion order:\( 3 \)
Leading coefficient: \( 2.965947 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(3\) \(3\) \(2\) \(3\) \(( 1 - T )( 1 + T )\)
\(11113\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 145 T + 11113 T^{2} )\)

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 \(\Q\).