Properties

 Label 990.a.240570.1 Conductor 990 Discriminant 240570 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![0, 3, 28, 72, 28, 3], R![0, 1, 1]);
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 3, 28, 72, 28, 3]), R([0, 1, 1]))

$y^2 + (x^2 + x)y = 3x^5 + 28x^4 + 72x^3 + 28x^2 + 3x$

Invariants

 magma: Conductor(LSeries(C)); Factorization($1); $$N$$ = $$990$$ = $$2 \cdot 3^{2} \cdot 5 \cdot 11$$ magma: Discriminant(C); Factorization(Integers()!$1); $$\Delta$$ = $$240570$$ = $$2 \cdot 3^{7} \cdot 5 \cdot 11$$

G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$306056$$ = $$2^{3} \cdot 67 \cdot 571$$ $$I_4$$ = $$27393028$$ = $$2^{2} \cdot 13 \cdot 263 \cdot 2003$$ $$I_6$$ = $$2746933168904$$ = $$2^{3} \cdot 31 \cdot 11076343423$$ $$I_{10}$$ = $$985374720$$ = $$2^{13} \cdot 3^{7} \cdot 5 \cdot 11$$ $$J_2$$ = $$38257$$ = $$67 \cdot 571$$ $$J_4$$ = $$60697908$$ = $$2^{2} \cdot 3^{2} \cdot 37 \cdot 45569$$ $$J_6$$ = $$127876480380$$ = $$2^{2} \cdot 3^{5} \cdot 5 \cdot 11 \cdot 2392003$$ $$J_8$$ = $$301983618580299$$ = $$3^{4} \cdot 19^{2} \cdot 20117 \cdot 513367$$ $$J_{10}$$ = $$240570$$ = $$2 \cdot 3^{7} \cdot 5 \cdot 11$$ $$g_1$$ = $$81951056110393451083057/240570$$ $$g_2$$ = $$188813894774599018858/13365$$ $$g_3$$ = $$7001861848004294/9$$
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 everywhere.

magma: [C![0,0,1],C![1,0,0]];

All rational points: (0 : 0 : 1), (1 : 0 : 0)

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

Number of rational Weierstrass points: $$2$$

Invariants of the Jacobian:

Analytic rank: $$0$$

magma: TwoSelmerGroup(Jacobian(C)); NumberOfGenerators($1); 2-Selmer rank: $$2$$ magma: HasSquareSha(Jacobian(C)); Order of Ш*: square Tamagawa numbers: 1 (p = 2), 2 (p = 3), 1 (p = 5), 1 (p = 11) magma: TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);

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

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 15.a2
Elliptic curve 66.b2

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 $$\Q$$.