Properties

Label 830.a.830000.1
Conductor 830
Discriminant -830000
Mordell-Weil group \(\Z/{2}\Z \times \Z/{8}\Z\)
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![0, 1, 8, 16, -2, 1], R![0, 1, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 1, 8, 16, -2, 1]), R([0, 1, 1]))
 

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 1, 8, 16, -2, 1], R![0, 1, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 4, 33, 66, -7, 4]))
 

$y^2 + (x^2 + x)y = x^5 - 2x^4 + 16x^3 + 8x^2 + x$ (homogenize, simplify)
$y^2 + (x^2z + xz^2)y = x^5z - 2x^4z^2 + 16x^3z^3 + 8x^2z^4 + xz^5$ (dehomogenize, simplify)
$y^2 = 4x^5 - 7x^4 + 66x^3 + 33x^2 + 4x$ (minimize, homogenize)

Invariants

\( N \)  =  \(830\) = \( 2 \cdot 5 \cdot 83 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
\( \Delta \)  =  \(-830000\) = \( - 2^{4} \cdot 5^{4} \cdot 83 \)
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);
 

\( I_2 \)  = \(30472\) =  \( 2^{3} \cdot 13 \cdot 293 \)
\( I_4 \)  = \(-917948\) =  \( - 2^{2} \cdot 229487 \)
\( I_6 \)  = \(-9181166648\) =  \( - 2^{3} \cdot 5779 \cdot 198589 \)
\( I_{10} \)  = \(-3399680000\) =  \( - 2^{16} \cdot 5^{4} \cdot 83 \)
\( J_2 \)  = \(3809\) =  \( 13 \cdot 293 \)
\( J_4 \)  = \(614082\) =  \( 2 \cdot 3 \cdot 7 \cdot 14621 \)
\( J_6 \)  = \(133745600\) =  \( 2^{6} \cdot 5^{2} \cdot 83591 \)
\( J_8 \)  = \(33085071919\) =  \( 33085071919 \)
\( J_{10} \)  = \(-830000\) =  \( - 2^{4} \cdot 5^{4} \cdot 83 \)
\( g_1 \)  = \(-801779343712318049/830000\)
\( g_2 \)  = \(-16967946642572289/415000\)
\( g_3 \)  = \(-4851113741084/2075\)

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![-1,6,4],C![0,0,1],C![1,-6,1],C![1,0,0],C![1,4,1]];
 

Points: \((0 : 0 : 1),\, (1 : 0 : 0),\, (1 : 4 : 1),\, (1 : -6 : 1),\, (-1 : 6 : 4)\)

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

Number of rational Weierstrass points: \(3\)

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: \(\Z/{2}\Z \times \Z/{8}\Z\)

Generator Height Order
\(4x + z\) \(=\) \(0,\) \(32y\) \(=\) \(3z^3\) \(0\) \(2\)
\(x^2 - 17xz - 4z^2\) \(=\) \(0,\) \(y\) \(=\) \(60xz^2 + 14z^3\) \(0\) \(8\)

2-torsion field: 3.1.83.1

BSD invariants

Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(2\)
Regulator: \( 1 \)
Real period: \( 5.868729 \)
Tamagawa product: \( 16 \)
Torsion order:\( 16 \)
Leading coefficient: \( 0.366795 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(2\) \(4\) \(1\) \(4\) \(( 1 - T )( 1 + T + 2 T^{2} )\)
\(5\) \(4\) \(1\) \(4\) \(( 1 - T )( 1 + 2 T + 5 T^{2} )\)
\(83\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 - 4 T + 83 T^{2} )\)

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 \(\Q\).