Show commands for: Magma / SageMath

The Jacobian of this curve is isogenous to that of the quotient of the modular curve $X_0(30)$ by the involution $W_2$ (see [MR:1373390]), which has discriminant $2\cdot 3^9\cdot 5^5$.

Minimal equation

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

$y^2 + (x^3 + 1)y = x^5 + 3x^4 + 3x^3 + 3x^2 + x$

Invariants

 magma: Conductor(LSeries(C)); Factorization($1); $$N$$ = $$450$$ = $$2 \cdot 3^{2} \cdot 5^{2}$$ magma: Discriminant(C); Factorization(Integers()!$1); $$\Delta$$ = $$-2700$$ = $$-1 \cdot 2^{2} \cdot 3^{3} \cdot 5^{2}$$

G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$-728$$ = $$-1 \cdot 2^{3} \cdot 7 \cdot 13$$ $$I_4$$ = $$14116$$ = $$2^{2} \cdot 3529$$ $$I_6$$ = $$-3145688$$ = $$-1 \cdot 2^{3} \cdot 7 \cdot 13 \cdot 29 \cdot 149$$ $$I_{10}$$ = $$-11059200$$ = $$-1 \cdot 2^{14} \cdot 3^{3} \cdot 5^{2}$$ $$J_2$$ = $$-91$$ = $$-1 \cdot 7 \cdot 13$$ $$J_4$$ = $$198$$ = $$2 \cdot 3^{2} \cdot 11$$ $$J_6$$ = $$0$$ = $$0$$ $$J_8$$ = $$-9801$$ = $$-1 \cdot 3^{4} \cdot 11^{2}$$ $$J_{10}$$ = $$-2700$$ = $$-1 \cdot 2^{2} \cdot 3^{3} \cdot 5^{2}$$ $$g_1$$ = $$6240321451/2700$$ $$g_2$$ = $$8289281/150$$ $$g_3$$ = $$0$$
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![-1,-1,1],C![-1,1,1],C![0,-1,1],C![0,0,1],C![1,-1,0],C![1,0,0]];

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

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*: square Tamagawa numbers: 2 (p = 2), 3 (p = 3), 1 (p = 5) magma: TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);

Torsion: $$\Z/{24}\Z$$

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.a7
Elliptic curve 30.a8

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 $$\Q$$.