Properties

 Label 55112.a.110224.1 Conductor 55112 Discriminant -110224 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![1, 0, 1, 0, 1], R![0, 1, 0, 1]);

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

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

Invariants

 magma: Conductor(LSeries(C)); Factorization($1); $$N$$ = $$55112$$ = $$2^{3} \cdot 83^{2}$$ magma: Discriminant(C); Factorization(Integers()!$1); $$\Delta$$ = $$-110224$$ = $$-1 \cdot 2^{4} \cdot 83^{2}$$

G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$-1440$$ = $$-1 \cdot 2^{5} \cdot 3^{2} \cdot 5$$ $$I_4$$ = $$16512$$ = $$2^{7} \cdot 3 \cdot 43$$ $$I_6$$ = $$-2002176$$ = $$-1 \cdot 2^{8} \cdot 3^{2} \cdot 11 \cdot 79$$ $$I_{10}$$ = $$-451477504$$ = $$-1 \cdot 2^{16} \cdot 83^{2}$$ $$J_2$$ = $$-180$$ = $$-1 \cdot 2^{2} \cdot 3^{2} \cdot 5$$ $$J_4$$ = $$1178$$ = $$2 \cdot 19 \cdot 31$$ $$J_6$$ = $$-18624$$ = $$-1 \cdot 2^{6} \cdot 3 \cdot 97$$ $$J_8$$ = $$491159$$ = $$491159$$ $$J_{10}$$ = $$-110224$$ = $$-1 \cdot 2^{4} \cdot 83^{2}$$ $$g_1$$ = $$11809800000/6889$$ $$g_2$$ = $$429381000/6889$$ $$g_3$$ = $$37713600/6889$$
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,-7,2],C![-1,-1,1],C![-1,3,1],C![-1,12,2],C![0,-1,1],C![0,1,1],C![1,-12,2],C![1,-3,1],C![1,-1,0],C![1,0,0],C![1,1,1],C![1,7,2]];

Known rational points: (-1 : -7 : 2), (-1 : -1 : 1), (-1 : 3 : 1), (-1 : 12 : 2), (0 : -1 : 1), (0 : 1 : 1), (1 : -12 : 2), (1 : -3 : 1), (1 : -1 : 0), (1 : 0 : 0), (1 : 1 : 1), (1 : 7 : 2)

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: $$3$$ magma: HasSquareSha(Jacobian(C)); Order of Ш*: square Regulator: 0.0342798375416 Real period: 11.991605395835409277223840169 Tamagawa numbers: 2 (p = 2), 1 (p = 83) 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 664.a1
Elliptic curve 83.a1

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 $$\Q$$.