Properties

Label 100096.c.100096.1
Conductor 100096
Discriminant -100096
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: Magma / SageMath

Minimal equation

Minimal equation

Simplified equation

$y^2 = x^5 - 9x^3 - 5x^2 + 21x + 20$ (homogenize, simplify)
$y^2 = x^5z - 9x^3z^3 - 5x^2z^4 + 21xz^5 + 20z^6$ (dehomogenize, simplify)
$y^2 = x^5 - 9x^3 - 5x^2 + 21x + 20$ (minimize, homogenize)

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![20, 21, -5, -9, 0, 1], R![]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([20, 21, -5, -9, 0, 1]), R([]));
 
magma: X,pi:= SimplifiedModel(C);
 
sage: X = HyperellipticCurve(R([20, 21, -5, -9, 0, 1]))
 

Invariants

Conductor: \( N \)  =  \(100096\) = \( 2^{8} \cdot 17 \cdot 23 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(-100096\) = \( - 2^{8} \cdot 17 \cdot 23 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(21216\) =  \( 2^{5} \cdot 3 \cdot 13 \cdot 17 \)
\( I_4 \)  = \(-344064\) =  \( - 2^{14} \cdot 3 \cdot 7 \)
\( I_6 \)  = \(-2446712832\) =  \( - 2^{13} \cdot 3 \cdot 29 \cdot 3433 \)
\( I_{10} \)  = \(-409993216\) =  \( - 2^{20} \cdot 17 \cdot 23 \)
\( J_2 \)  = \(2652\) =  \( 2^{2} \cdot 3 \cdot 13 \cdot 17 \)
\( J_4 \)  = \(296630\) =  \( 2 \cdot 5 \cdot 29663 \)
\( J_6 \)  = \(44782996\) =  \( 2^{2} \cdot 101 \cdot 110849 \)
\( J_8 \)  = \(7693787123\) =  \( 11 \cdot 461 \cdot 1517213 \)
\( J_{10} \)  = \(-100096\) =  \( - 2^{8} \cdot 17 \cdot 23 \)
\( g_1 \)  = \(-30142461298716/23\)
\( g_2 \)  = \(-2542592373165/46\)
\( g_3 \)  = \(-289488481893/92\)

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

magma: [C![1,0,0]];
 

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

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(D_0 - 2 \cdot(1 : 0 : 0)\) \(x^2 - xz - 4z^2\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(2\)

2-torsion field: 6.2.2598977.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(1\)
Regulator: \( 1 \)
Real period: \( 11.27396 \)
Tamagawa product: \( 1 \)
Torsion order:\( 2 \)
Leading coefficient: \( 2.818491 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(2\) \(8\) \(8\) \(1\) \(1\)
\(17\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 4 T + 17 T^{2} )\)
\(23\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 - 2 T + 23 T^{2} )\)

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 \(\Q\).