Properties

Label 100309.a.100309.1
Conductor 100309
Discriminant 100309
Mordell-Weil group \(\Z/{2}\Z\)
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: SageMath / Magma

Minimal equation

Minimal equation

Simplified equation

$y^2 + xy = x^5 + 2x^3 - 3x^2 + x$ (homogenize, simplify)
$y^2 + xz^2y = x^5z + 2x^3z^3 - 3x^2z^4 + xz^5$ (dehomogenize, simplify)
$y^2 = 4x^5 + 8x^3 - 11x^2 + 4x$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 1, -3, 2, 0, 1]), R([0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 1, -3, 2, 0, 1], R![0, 1]);
 
sage: X = HyperellipticCurve(R([0, 4, -11, 8, 0, 4]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  =  \(100309\) = \( 11^{2} \cdot 829 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(100309\) = \( 11^{2} \cdot 829 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(1024\) =  \( 2^{10} \)
\( I_4 \)  = \(-30080\) =  \( - 2^{7} \cdot 5 \cdot 47 \)
\( I_6 \)  = \(-6478400\) =  \( - 2^{6} \cdot 5^{2} \cdot 4049 \)
\( I_{10} \)  = \(410865664\) =  \( 2^{12} \cdot 11^{2} \cdot 829 \)
\( J_2 \)  = \(128\) =  \( 2^{7} \)
\( J_4 \)  = \(996\) =  \( 2^{2} \cdot 3 \cdot 83 \)
\( J_6 \)  = \(4961\) =  \( 11^{2} \cdot 41 \)
\( J_8 \)  = \(-89252\) =  \( - 2^{2} \cdot 53 \cdot 421 \)
\( J_{10} \)  = \(100309\) =  \( 11^{2} \cdot 829 \)
\( g_1 \)  = \(34359738368/100309\)
\( g_2 \)  = \(2088763392/100309\)
\( g_3 \)  = \(671744/829\)

sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]
 
magma: IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);
 

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: \((1 : 0 : 0),\, (0 : 0 : 1)\)

magma: [C![0,0,1],C![1,0,0]];
 

Number of rational Weierstrass points: \(2\)

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: \(\Z/{2}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((0 : 0 : 1) - (1 : 0 : 0)\) \(x\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(2\)

2-torsion field: 4.0.401236.2

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(1\)
Regulator: \( 1 \)
Real period: \( 10.47180 \)
Tamagawa product: \( 1 \)
Torsion order:\( 2 \)
Leading coefficient: \( 2.617950 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(11\) \(2\) \(2\) \(1\) \(( 1 - T )( 1 + T )\)
\(829\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 34 T + 829 T^{2} )\)

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 \(\Q\).