Properties

 Label 70450.c.704500.1 Conductor 70450 Discriminant -704500 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, -3, 4, 1, -2], R![1, 0, 0, 1]);

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -3, 4, 1, -2]), R([1, 0, 0, 1]))

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

Invariants

 magma: Conductor(LSeries(C)); Factorization($1); $$N$$ = $$70450$$ = $$2 \cdot 5^{2} \cdot 1409$$ magma: Discriminant(C); Factorization(Integers()!$1); $$\Delta$$ = $$-704500$$ = $$-1 \cdot 2^{2} \cdot 5^{3} \cdot 1409$$

G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$2024$$ = $$2^{3} \cdot 11 \cdot 23$$ $$I_4$$ = $$232420$$ = $$2^{2} \cdot 5 \cdot 11621$$ $$I_6$$ = $$107744360$$ = $$2^{3} \cdot 5 \cdot 2693609$$ $$I_{10}$$ = $$-2885632000$$ = $$-1 \cdot 2^{14} \cdot 5^{3} \cdot 1409$$ $$J_2$$ = $$253$$ = $$11 \cdot 23$$ $$J_4$$ = $$246$$ = $$2 \cdot 3 \cdot 41$$ $$J_6$$ = $$20576$$ = $$2^{5} \cdot 643$$ $$J_8$$ = $$1286303$$ = $$1286303$$ $$J_{10}$$ = $$-704500$$ = $$-1 \cdot 2^{2} \cdot 5^{3} \cdot 1409$$ $$g_1$$ = $$-1036579476493/704500$$ $$g_2$$ = $$-1991896071/352250$$ $$g_3$$ = $$-329262296/176125$$
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![-3,0,2],C![-3,8,1],C![-3,18,1],C![-3,19,2],C![-1,-16,2],C![-1,-2,1],C![-1,2,1],C![-1,9,2],C![0,-1,1],C![0,0,1],C![1,-2,1],C![1,-1,0],C![1,0,0],C![1,0,1],C![2,-26,3],C![2,-9,3],C![2,-7,1],C![2,-2,1],C![10,-882,3],C![10,-145,3],C![29,-23200,5],C![29,-1314,5]];

Known rational points: (-3 : 0 : 2), (-3 : 8 : 1), (-3 : 18 : 1), (-3 : 19 : 2), (-1 : -16 : 2), (-1 : -2 : 1), (-1 : 2 : 1), (-1 : 9 : 2), (0 : -1 : 1), (0 : 0 : 1), (1 : -2 : 1), (1 : -1 : 0), (1 : 0 : 0), (1 : 0 : 1), (2 : -26 : 3), (2 : -9 : 3), (2 : -7 : 1), (2 : -2 : 1), (10 : -882 : 3), (10 : -145 : 3), (29 : -23200 : 5), (29 : -1314 : 5)

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: $$4$$ magma: HasSquareSha(Jacobian(C)); Order of Ш*: square Regulator: 0.046457640618 Real period: 16.521293553936620171892200688 Tamagawa numbers: 2 (p = 2), 2 (p = 5), 1 (p = 1409) magma: TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);

Torsion: $$\Z/{2}\Z$$

2-torsion field: splitting field of $$x^{6} - 17 x^{4} - 64 x^{3} - 174 x^{2} - 236 x - 56$$ with Galois group $S_4\times C_2$

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 $$\Q$$.