Properties

 Label 1083.b.390963.1 Conductor 1083 Discriminant -390963 Sato-Tate group $G_{3,3}$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R \times \R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\Q \times \Q$$ $$\overline{\Q}$$-simple no $$\mathrm{GL}_2$$-type yes

Related objects

Show commands for: Magma / SageMath

Minimal equation

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-6, -33, -14, 95, -50, 3, -1], R![1]);
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-6, -33, -14, 95, -50, 3, -1]), R([1]))

$y^2 + y = -x^6 + 3x^5 - 50x^4 + 95x^3 - 14x^2 - 33x - 6$

Invariants

 magma: Conductor(LSeries(C)); Factorization($1); $$N$$ = $$1083$$ = $$3 \cdot 19^{2}$$ magma: Discriminant(C); Factorization(Integers()!$1); $$\Delta$$ = $$-390963$$ = $$-1 \cdot 3 \cdot 19^{4}$$

G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$601760$$ = $$2^{5} \cdot 5 \cdot 3761$$ $$I_4$$ = $$31128254272$$ = $$2^{6} \cdot 486378973$$ $$I_6$$ = $$4413239365215232$$ = $$2^{10} \cdot 17 \cdot 97 \cdot 2857 \cdot 914801$$ $$I_{10}$$ = $$-1601384448$$ = $$-1 \cdot 2^{12} \cdot 3 \cdot 19^{4}$$ $$J_2$$ = $$75220$$ = $$2^{2} \cdot 5 \cdot 3761$$ $$J_4$$ = $$-88500632$$ = $$-1 \cdot 2^{3} \cdot 11 \cdot 19 \cdot 41 \cdot 1291$$ $$J_6$$ = $$98386538568$$ = $$2^{3} \cdot 3 \cdot 19^{3} \cdot 597673$$ $$J_8$$ = $$-107931608328616$$ = $$-1 \cdot 2^{3} \cdot 19^{2} \cdot 37 \cdot 1010065961$$ $$J_{10}$$ = $$-390963$$ = $$-1 \cdot 3 \cdot 19^{4}$$ $$g_1$$ = $$-2408056349828975363200000/390963$$ $$g_2$$ = $$1982406707133537344000/20577$$ $$g_3$$ = $$-27053302090985600/19$$
Alternative geometric invariants: G2

Automorphism group

 magma: AutomorphismGroup(C); IdentifyGroup($1); $$\mathrm{Aut}(X)$$ $$\simeq$$ $$V_4$$ (GAP id : [4,2]) magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1); $$\mathrm{Aut}(X_{\overline{\Q}})$$ $$\simeq$$ $$V_4$$ (GAP id : [4,2])

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 except over $\R$.

magma: [];

There are no rational points.

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

Number of rational Weierstrass points: $$0$$

Invariants of the Jacobian:

Analytic rank: $$0$$

magma: TwoSelmerGroup(Jacobian(C)); NumberOfGenerators($1); 2-Selmer rank: $$1$$ magma: HasSquareSha(Jacobian(C)); Order of Ш*: twice a square Tamagawa numbers: 1 (p = 3), 1 (p = 19) magma: TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);

Torsion: $$\mathrm{trivial}$$

Sato-Tate group

 $$\mathrm{ST}$$ $$\simeq$$ $G_{3,3}$ $$\mathrm{ST}^0$$ $$\simeq$$ $$\mathrm{SU}(2)\times\mathrm{SU}(2)$$

Decomposition

Splits over $$\Q$$

Decomposes up to isogeny as the product of the non-isogenous elliptic curves:
Elliptic curve 19.a1
Elliptic curve 57.b1

Endomorphisms

Of $$\GL_2$$-type over $$\Q$$

Endomorphism ring over $$\Q$$:
 $$\End (J_{})$$ $$\simeq$$ an order of index $$2$$ in $$\Z \times \Z$$ $$\End (J_{}) \otimes \Q$$ $$\simeq$$ $$\Q$$ $$\times$$ $$\Q$$ $$\End (J_{}) \otimes \R$$ $$\simeq$$ $$\R \times \R$$

All $$\overline{\Q}$$-endomorphisms of the Jacobian are defined over $$\Q$$.