Properties

Label 464.a.29696.2
Conductor 464
Discriminant -29696
Mordell-Weil group \(\Z/{2}\Z \times \Z/{2}\Z\)
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![0, 1, 16, 72, 33, 4], R![0, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 1, 16, 72, 33, 4]), R([0, 1]))
 

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 1, 16, 72, 33, 4], R![0, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 4, 65, 288, 132, 16]))
 

$y^2 + xy = 4x^5 + 33x^4 + 72x^3 + 16x^2 + x$ (homogenize, simplify)
$y^2 + xz^2y = 4x^5z + 33x^4z^2 + 72x^3z^3 + 16x^2z^4 + xz^5$ (dehomogenize, simplify)
$y^2 = 16x^5 + 132x^4 + 288x^3 + 65x^2 + 4x$ (minimize, homogenize)

Invariants

\( N \)  =  \(464\) = \( 2^{4} \cdot 29 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
\( \Delta \)  =  \(-29696\) = \( - 2^{10} \cdot 29 \)
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);
 

\( I_2 \)  = \(362944\) =  \( 2^{6} \cdot 53 \cdot 107 \)
\( I_4 \)  = \(12942400\) =  \( 2^{6} \cdot 5^{2} \cdot 8089 \)
\( I_6 \)  = \(1542241461760\) =  \( 2^{9} \cdot 5 \cdot 67 \cdot 8991613 \)
\( I_{10} \)  = \(-121634816\) =  \( - 2^{22} \cdot 29 \)
\( J_2 \)  = \(45368\) =  \( 2^{3} \cdot 53 \cdot 107 \)
\( J_4 \)  = \(85625826\) =  \( 2 \cdot 3 \cdot 11 \cdot 13 \cdot 23 \cdot 4339 \)
\( J_6 \)  = \(215176422416\) =  \( 2^{4} \cdot 19 \cdot 23557 \cdot 30047 \)
\( J_8 \)  = \(607585463496703\) =  \( 17 \cdot 727 \cdot 49161377417 \)
\( J_{10} \)  = \(-29696\) =  \( - 2^{10} \cdot 29 \)
\( g_1 \)  = \(-187693059992988715232/29\)
\( g_2 \)  = \(-7808250185554819143/29\)
\( g_3 \)  = \(-432507850151022641/29\)

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![-4,2,1],C![0,0,1],C![1,0,0]];
 

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

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

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

2-torsion field: 3.1.116.1

BSD invariants

Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(2\)
Regulator: \( 1 \)
Real period: \( 1.802678 \)
Tamagawa product: \( 2 \)
Torsion order:\( 4 \)
Leading coefficient: \( 0.225334 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(2\) \(10\) \(4\) \(2\) \(1 + T + 2 T^{2}\)
\(29\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 2 T + 29 T^{2} )\)

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 \(\Q\).