Properties

 Label 289238.a.289238.1 Conductor 289238 Discriminant -289238 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![10, 40, 51, 20, 0, 1, -1], R![1, 1, 1]);
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([10, 40, 51, 20, 0, 1, -1]), R([1, 1, 1]))

$y^2 + (x^2 + x + 1)y = -x^6 + x^5 + 20x^3 + 51x^2 + 40x + 10$

Invariants

 magma: Conductor(LSeries(C)); Factorization($1); $$N$$ = $$289238$$ = $$2 \cdot 17 \cdot 47 \cdot 181$$ magma: Discriminant(C); Factorization(Integers()!$1); $$\Delta$$ = $$-289238$$ = $$-1 \cdot 2 \cdot 17 \cdot 47 \cdot 181$$

G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$102312$$ = $$2^{3} \cdot 3^{2} \cdot 7^{2} \cdot 29$$ $$I_4$$ = $$-375420$$ = $$-1 \cdot 2^{2} \cdot 3 \cdot 5 \cdot 6257$$ $$I_6$$ = $$-12763734264$$ = $$-1 \cdot 2^{3} \cdot 3^{2} \cdot 23 \cdot 7707569$$ $$I_{10}$$ = $$-1184718848$$ = $$-1 \cdot 2^{13} \cdot 17 \cdot 47 \cdot 181$$ $$J_2$$ = $$12789$$ = $$3^{2} \cdot 7^{2} \cdot 29$$ $$J_4$$ = $$6818849$$ = $$6818849$$ $$J_6$$ = $$4850280481$$ = $$179 \cdot 431 \cdot 62869$$ $$J_8$$ = $$3883383846677$$ = $$31 \cdot 78941 \cdot 1586887$$ $$J_{10}$$ = $$-289238$$ = $$-1 \cdot 2 \cdot 17 \cdot 47 \cdot 181$$ $$g_1$$ = $$-342123524046146462949/289238$$ $$g_2$$ = $$-14263326884806825581/289238$$ $$g_3$$ = $$-793304701907528601/289238$$
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: [];

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

Number of rational Weierstrass points: $$0$$

Invariants of the Jacobian:

Analytic rank: $$1$$

magma: TwoSelmerGroup(Jacobian(C)); NumberOfGenerators($1); 2-Selmer rank: $$3$$ magma: HasSquareSha(Jacobian(C)); Order of Ш*: square Tamagawa numbers: 1 (p = 2), 1 (p = 17), 1 (p = 47), 1 (p = 181) magma: TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);

Torsion: $$\mathrm{trivial}$$

2-torsion field: splitting field of $$x^{6} - 6 x^{4} - 6 x^{3} - 2 x^{2} + 8 x + 11$$ with Galois group $S_6$

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 $$\Q$$.