Properties

 Label 49507.a.49507.1 Conductor 49507 Discriminant -49507 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

Related objects

Show commands for: Magma / SageMath

Minimal equation

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 1, 3, 3, 2], R![1, 0, 0, 1]);

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 1, 3, 3, 2]), R([1, 0, 0, 1]))

$y^2 + (x^3 + 1)y = 2x^4 + 3x^3 + 3x^2 + x$

Invariants

 magma: Conductor(LSeries(C)); Factorization($1); $$N$$ = $$49507$$ = $$31 \cdot 1597$$ magma: Discriminant(C); Factorization(Integers()!$1); $$\Delta$$ = $$-49507$$ = $$-1 \cdot 31 \cdot 1597$$

G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$-600$$ = $$-1 \cdot 2^{3} \cdot 3 \cdot 5^{2}$$ $$I_4$$ = $$35556$$ = $$2^{2} \cdot 3 \cdot 2963$$ $$I_6$$ = $$-4357272$$ = $$-1 \cdot 2^{3} \cdot 3 \cdot 181553$$ $$I_{10}$$ = $$-202780672$$ = $$-1 \cdot 2^{12} \cdot 31 \cdot 1597$$ $$J_2$$ = $$-75$$ = $$-1 \cdot 3 \cdot 5^{2}$$ $$J_4$$ = $$-136$$ = $$-1 \cdot 2^{3} \cdot 17$$ $$J_6$$ = $$-1128$$ = $$-1 \cdot 2^{3} \cdot 3 \cdot 47$$ $$J_8$$ = $$16526$$ = $$2 \cdot 8263$$ $$J_{10}$$ = $$-49507$$ = $$-1 \cdot 31 \cdot 1597$$ $$g_1$$ = $$2373046875/49507$$ $$g_2$$ = $$-57375000/49507$$ $$g_3$$ = $$6345000/49507$$
Alternative geometric invariants: G2

Automorphism group

 magma: AutomorphismGroup(C); IdentifyGroup($1); $$\mathrm{Aut}(X)$$ $$\simeq$$ $$C_2$$ (GAP id : [2,1]) magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1); $$\mathrm{Aut}(X_{\overline{\Q}})$$ $$\simeq$$ $$C_2$$ (GAP id : [2,1])

Rational points

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.

magma: [C![-12,-21,1],C![-12,1748,1],C![-2,-2,1],C![-2,9,1],C![-1,-7,2],C![-1,-1,1],C![-1,0,2],C![-1,1,1],C![0,-1,1],C![0,0,1],C![1,-16,2],C![1,-1,0],C![1,0,0],C![1,7,2],C![2,-14,1],C![2,5,1],C![9,-84825,40],C![9,20096,40]];

Known rational points: (-12 : -21 : 1), (-12 : 1748 : 1), (-2 : -2 : 1), (-2 : 9 : 1), (-1 : -7 : 2), (-1 : -1 : 1), (-1 : 0 : 2), (-1 : 1 : 1), (0 : -1 : 1), (0 : 0 : 1), (1 : -16 : 2), (1 : -1 : 0), (1 : 0 : 0), (1 : 7 : 2), (2 : -14 : 1), (2 : 5 : 1), (9 : -84825 : 40), (9 : 20096 : 40)

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

Number of rational Weierstrass points: $$0$$

Invariants of the Jacobian:

Analytic rank*: $$3$$

magma: TwoSelmerGroup(Jacobian(C)); NumberOfGenerators($1); 2-Selmer rank: $$3$$ magma: HasSquareSha(Jacobian(C)); Order of Ш*: square Regulator: 0.0407445836158 Real period: 15.858977223376220431511869621 Tamagawa numbers: 1 (p = 31), 1 (p = 1597) magma: TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);

Torsion: $$\mathrm{trivial}$$

Sato-Tate group

 $$\mathrm{ST}$$ $$\simeq$$ $\mathrm{USp}(4)$ $$\mathrm{ST}^0$$ $$\simeq$$ $$\mathrm{USp}(4)$$

Decomposition

Simple over $$\overline{\Q}$$

Endomorphisms

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 $$\Q$$.