Properties

Label 100224.a.601344.1
Conductor 100224
Discriminant 601344
Mordell-Weil group \(\Z \times \Z/{2}\Z\)
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

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Minimal equation

Minimal equation

Simplified equation

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-1, 1, 2, 4, 2, 1], R![]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-1, 1, 2, 4, 2, 1]), R([]))
 

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-1, 1, 2, 4, 2, 1], R![]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-1, 1, 2, 4, 2, 1]))
 

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

Invariants

\( N \)  =  \(100224\) = \( 2^{7} \cdot 3^{3} \cdot 29 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
\( \Delta \)  =  \(601344\) = \( 2^{8} \cdot 3^{4} \cdot 29 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

magma: IgusaClebschInvariants(C); [Factorization(Integers()!a): a in $1];
 
sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]
 

Igusa invariants

magma: IgusaInvariants(C); [Factorization(Integers()!a): a in $1];
 

G2 invariants

magma: G2Invariants(C);
 

\( I_2 \)  = \(1152\) =  \( 2^{7} \cdot 3^{2} \)
\( I_4 \)  = \(165888\) =  \( 2^{11} \cdot 3^{4} \)
\( I_6 \)  = \(72843264\) =  \( 2^{15} \cdot 3^{2} \cdot 13 \cdot 19 \)
\( I_{10} \)  = \(2463105024\) =  \( 2^{20} \cdot 3^{4} \cdot 29 \)
\( J_2 \)  = \(144\) =  \( 2^{4} \cdot 3^{2} \)
\( J_4 \)  = \(-864\) =  \( - 2^{5} \cdot 3^{3} \)
\( J_6 \)  = \(-50432\) =  \( - 2^{8} \cdot 197 \)
\( J_8 \)  = \(-2002176\) =  \( - 2^{8} \cdot 3^{2} \cdot 11 \cdot 79 \)
\( J_{10} \)  = \(601344\) =  \( 2^{8} \cdot 3^{4} \cdot 29 \)
\( g_1 \)  = \(2985984/29\)
\( g_2 \)  = \(-124416/29\)
\( g_3 \)  = \(-50432/29\)

Automorphism group

magma: AutomorphismGroup(C); IdentifyGroup($1);
 
\(\mathrm{Aut}(X)\)\(\simeq\) $C_2$
magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
 
\(\mathrm{Aut}(X_{\overline{\Q}})\)\(\simeq\) $C_2$

Rational points

magma: [C![1,-3,1],C![1,0,0],C![1,3,1]];
 

Points: \((1 : 0 : 0),\, (1 : -3 : 1),\, (1 : 3 : 1)\)

magma: #Roots(HyperellipticPolynomials(SimplifiedModel(C)));
 

Number of rational Weierstrass points: \(1\)

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);
 

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian:

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Group structure: \(\Z \times \Z/{2}\Z\)

Generator Height Order
\(x - z\) \(=\) \(0,\) \(y\) \(=\) \(-3z^3\) \(0.501426\) \(\infty\)
\(x^2 + xz + z^2\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0\) \(2\)

2-torsion field: 6.0.22707.1

BSD invariants

Analytic rank: \(1\)   (upper bound)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(2\)
Regulator: \( 0.501426 \)
Real period: \( 5.342534 \)
Tamagawa product: \( 4 \)
Torsion order:\( 2 \)
Leading coefficient: \( 2.678888 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(2\) \(8\) \(7\) \(2\) \(1\)
\(3\) \(4\) \(3\) \(2\) \(1 - T\)
\(29\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 4 T + 29 T^{2} )\)

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 \(\Q\).