Properties

Label 461.a.461.2
Conductor $461$
Discriminant $461$
Mordell-Weil group trivial
Sato-Tate group $\mathrm{USp}(4)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q\)
\(\End(J) \otimes \Q\) \(\Q\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type no

Related objects

Downloads

Learn more

Show commands: SageMath / Magma

Minimal equation

Minimal equation

Simplified equation

$y^2 + y = x^5 - x^4 - 39x^3 + 10x^2 + 272x - 306$ (homogenize, simplify)
$y^2 + z^3y = x^5z - x^4z^2 - 39x^3z^3 + 10x^2z^4 + 272xz^5 - 306z^6$ (dehomogenize, simplify)
$y^2 = 4x^5 - 4x^4 - 156x^3 + 40x^2 + 1088x - 1223$ (homogenize, minimize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-306, 272, 10, -39, -1, 1]), R([1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-306, 272, 10, -39, -1, 1], R![1]);
 
sage: X = HyperellipticCurve(R([-1223, 1088, 40, -156, -4, 4]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(461\) \(=\) \( 461 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(461\) \(=\) \( 461 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(80664\) \(=\)  \( 2^{3} \cdot 3 \cdot 3361 \)
\( I_4 \)  \(=\) \(166117104\) \(=\)  \( 2^{4} \cdot 3^{2} \cdot 29 \cdot 39779 \)
\( I_6 \)  \(=\) \(3752725952952\) \(=\)  \( 2^{3} \cdot 3^{2} \cdot 71 \cdot 1319 \cdot 556559 \)
\( I_{10} \)  \(=\) \(1844\) \(=\)  \( 2^{2} \cdot 461 \)
\( J_2 \)  \(=\) \(40332\) \(=\)  \( 2^{2} \cdot 3 \cdot 3361 \)
\( J_4 \)  \(=\) \(40091742\) \(=\)  \( 2 \cdot 3^{2} \cdot 31 \cdot 71849 \)
\( J_6 \)  \(=\) \(45075737276\) \(=\)  \( 2^{2} \cdot 83 \cdot 135770293 \)
\( J_8 \)  \(=\) \(52661714805267\) \(=\)  \( 3 \cdot 613 \cdot 8819 \cdot 3247087 \)
\( J_{10} \)  \(=\) \(461\) \(=\)  \( 461 \)
\( g_1 \)  \(=\) \(106720731303787612818432/461\)
\( g_2 \)  \(=\) \(2630293443843585469056/461\)
\( g_3 \)  \(=\) \(73323359651716069824/461\)

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: \((1 : 0 : 0)\)
All points: \((1 : 0 : 0)\)
All points: \((1 : 0 : 0)\)

magma: [C![1,0,0]]; // minimal model
 
magma: [C![1,0,0]]; // simplified model
 

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: 5.1.7376.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(0\)
Regulator: \( 1 \)
Real period: \( 0.245886 \)
Tamagawa product: \( 1 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.245886 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(461\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 + 461 T^{2} )\)

Galois representations

For primes $\ell \le 3$, the image of the mod-$\ell$ Galois representation is listed in the table below, whenever it is not the maximal image $\GSp(4,\F_\ell)$

Prime \(\ell\) mod-\(\ell\) image
\(2\) 2.6.1

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 \(\Q\).