Properties

Label 100000.a.200000.1
Conductor 100000
Discriminant 200000
Mordell-Weil group trivial
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

$y^2 + y = 2x^5 + 5x^3 - 10x^2 + 5x - 1$ (homogenize, simplify)
$y^2 + z^3y = 2x^5z + 5x^3z^3 - 10x^2z^4 + 5xz^5 - z^6$ (dehomogenize, simplify)
$y^2 = 8x^5 + 20x^3 - 40x^2 + 20x - 3$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-1, 5, -10, 5, 0, 2]), R([1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-1, 5, -10, 5, 0, 2], R![1]);
 
sage: X = HyperellipticCurve(R([-3, 20, -40, 20, 0, 8]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  =  \(100000\) = \( 2^{5} \cdot 5^{5} \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(200000\) = \( 2^{6} \cdot 5^{5} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(8800\) =  \( 2^{5} \cdot 5^{2} \cdot 11 \)
\( I_4 \)  = \(-512000\) =  \( - 2^{12} \cdot 5^{3} \)
\( I_6 \)  = \(-1768960000\) =  \( - 2^{12} \cdot 5^{4} \cdot 691 \)
\( I_{10} \)  = \(819200000\) =  \( 2^{18} \cdot 5^{5} \)
\( J_2 \)  = \(1100\) =  \( 2^{2} \cdot 5^{2} \cdot 11 \)
\( J_4 \)  = \(55750\) =  \( 2 \cdot 5^{3} \cdot 223 \)
\( J_6 \)  = \(4522500\) =  \( 2^{2} \cdot 3^{3} \cdot 5^{4} \cdot 67 \)
\( J_8 \)  = \(466671875\) =  \( 5^{6} \cdot 29867 \)
\( J_{10} \)  = \(200000\) =  \( 2^{6} \cdot 5^{5} \)
\( g_1 \)  = \(8052550000\)
\( g_2 \)  = \(371016250\)
\( g_3 \)  = \(27361125\)

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

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: trivial

magma: MordellWeilGroupGenus2(Jacobian(C));
 

2-torsion field: 5.1.50000.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(0\)
Regulator: \( 1 \)
Real period: \( 2.393619 \)
Tamagawa product: \( 2 \)
Torsion order:\( 1 \)
Leading coefficient: \( 4.787238 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(2\) \(6\) \(5\) \(2\) \(1 - 2 T + 2 T^{2}\)
\(5\) \(5\) \(5\) \(1\) \(1\)

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 \(\Q\).