Properties

Label 604.a.9664.1
Conductor 604
Discriminant 9664
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

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Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^2 + x + 1)y = 4x^5 + 9x^4 + 48x^3 - 4x^2 - 53x - 21$ (homogenize, simplify)
$y^2 + (x^2z + xz^2 + z^3)y = 4x^5z + 9x^4z^2 + 48x^3z^3 - 4x^2z^4 - 53xz^5 - 21z^6$ (dehomogenize, simplify)
$y^2 = 16x^5 + 37x^4 + 194x^3 - 13x^2 - 210x - 83$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-21, -53, -4, 48, 9, 4]), R([1, 1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-21, -53, -4, 48, 9, 4], R![1, 1, 1]);
 
sage: X = HyperellipticCurve(R([-83, -210, -13, 194, 37, 16]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  =  \(604\) = \( 2^{2} \cdot 151 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(9664\) = \( 2^{6} \cdot 151 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(99112\) =  \( 2^{3} \cdot 13 \cdot 953 \)
\( I_4 \)  = \(-3188351900\) =  \( - 2^{2} \cdot 5^{2} \cdot 457 \cdot 69767 \)
\( I_6 \)  = \(-191974986700824\) =  \( - 2^{3} \cdot 3 \cdot 20747 \cdot 385547683 \)
\( I_{10} \)  = \(39583744\) =  \( 2^{18} \cdot 151 \)
\( J_2 \)  = \(12389\) =  \( 13 \cdot 953 \)
\( J_4 \)  = \(39607304\) =  \( 2^{3} \cdot 11 \cdot 450083 \)
\( J_6 \)  = \(223396249616\) =  \( 2^{4} \cdot 10289 \cdot 1357009 \)
\( J_8 \)  = \(299729401586052\) =  \( 2^{2} \cdot 3 \cdot 2971 \cdot 8407085201 \)
\( J_{10} \)  = \(9664\) =  \( 2^{6} \cdot 151 \)
\( g_1 \)  = \(291864493641401980949/9664\)
\( g_2 \)  = \(9414430497536890397/1208\)
\( g_3 \)  = \(2143030742187944921/604\)

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: 5.1.9664.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(0\)
Regulator: \( 1 \)
Real period: \( 0.291788 \)
Tamagawa product: \( 1 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.291788 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(2\) \(6\) \(2\) \(1\) \(( 1 - T )( 1 + T )\)
\(151\) \(1\) \(1\) \(1\) \(( 1 - T )( 1 + 10 T + 151 T^{2} )\)

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 \(\Q\).