Properties

Label 100162.b.400648.1
Conductor 100162
Discriminant -400648
Mordell-Weil group \(\Z \times \Z/{3}\Z\)
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

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Minimal equation

Minimal equation

Simplified equation

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![2, 5, 4, 0, -4, 1], R![0, 1, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([2, 5, 4, 0, -4, 1]), R([0, 1, 1]))
 

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![2, 5, 4, 0, -4, 1], R![0, 1, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([8, 20, 17, 2, -15, 4]))
 

$y^2 + (x^2 + x)y = x^5 - 4x^4 + 4x^2 + 5x + 2$ (homogenize, simplify)
$y^2 + (x^2z + xz^2)y = x^5z - 4x^4z^2 + 4x^2z^4 + 5xz^5 + 2z^6$ (dehomogenize, simplify)
$y^2 = 4x^5 - 15x^4 + 2x^3 + 17x^2 + 20x + 8$ (minimize, homogenize)

Invariants

\( N \)  =  \(100162\) = \( 2 \cdot 61 \cdot 821 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
\( \Delta \)  =  \(-400648\) = \( - 2^{3} \cdot 61 \cdot 821 \)
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);
 

\( I_2 \)  = \(7304\) =  \( 2^{3} \cdot 11 \cdot 83 \)
\( I_4 \)  = \(-928124\) =  \( - 2^{2} \cdot 331 \cdot 701 \)
\( I_6 \)  = \(-2399465848\) =  \( - 2^{3} \cdot 13 \cdot 23071787 \)
\( I_{10} \)  = \(-1641054208\) =  \( - 2^{15} \cdot 61 \cdot 821 \)
\( J_2 \)  = \(913\) =  \( 11 \cdot 83 \)
\( J_4 \)  = \(44400\) =  \( 2^{4} \cdot 3 \cdot 5^{2} \cdot 37 \)
\( J_6 \)  = \(3475524\) =  \( 2^{2} \cdot 3 \cdot 13 \cdot 22279 \)
\( J_8 \)  = \(300448353\) =  \( 3 \cdot 43 \cdot 293 \cdot 7949 \)
\( J_{10} \)  = \(-400648\) =  \( - 2^{3} \cdot 61 \cdot 821 \)
\( g_1 \)  = \(-634386434595793/400648\)
\( g_2 \)  = \(-4223819158350/50081\)
\( g_3 \)  = \(-724272266289/100162\)

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,-4,1],C![1,0,0],C![1,2,1]];
 

Points: \((1 : 0 : 0),\, (1 : 2 : 1),\, (1 : -4 : 1)\)

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

Generator Height Order
\(x - z\) \(=\) \(0,\) \(y\) \(=\) \(-4z^3\) \(0.772491\) \(\infty\)
\(x^2 - 2xz - z^2\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2 - z^3\) \(0\) \(3\)

2-torsion field: 5.3.1602592.1

BSD invariants

Analytic rank: \(1\)   (upper bound)
Mordell-Weil rank: \(1\)
2-Selmer rank:\(1\)
Regulator: \( 0.772491 \)
Real period: \( 11.20513 \)
Tamagawa product: \( 3 \)
Torsion order:\( 3 \)
Leading coefficient: \( 2.885288 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(2\) \(3\) \(1\) \(3\) \(( 1 - T )( 1 + T + 2 T^{2} )\)
\(61\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 13 T + 61 T^{2} )\)
\(821\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 - 21 T + 821 T^{2} )\)

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 \(\Q\).