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

Show commands: SageMath / Magma

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);

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));

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 $$\Q$$.