# Properties

 Label 1088.a.1088.1 Conductor 1088 Discriminant -1088 Sato-Tate group $N(G_{3,3})$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R \times \R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\Q \times \Q$$ $$\overline{\Q}$$-simple no $$\mathrm{GL}_2$$-type no

# Related objects

Show commands for: Magma / SageMath

## Minimal equation

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![1, 1, 2, 1, 1], R![1, 1, 1, 1]);
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([1, 1, 2, 1, 1]), R([1, 1, 1, 1]))

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

## Invariants

 magma: Conductor(LSeries(C)); Factorization($1); $$N$$ = $$1088$$ = $$2^{6} \cdot 17$$ magma: Discriminant(C); Factorization(Integers()!$1); $$\Delta$$ = $$-1088$$ = $$-1 \cdot 2^{6} \cdot 17$$

### G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$-1568$$ = $$-1 \cdot 2^{5} \cdot 7^{2}$$ $$I_4$$ = $$1792$$ = $$2^{8} \cdot 7$$ $$I_6$$ = $$-323584$$ = $$-1 \cdot 2^{12} \cdot 79$$ $$I_{10}$$ = $$-4456448$$ = $$-1 \cdot 2^{18} \cdot 17$$ $$J_2$$ = $$-196$$ = $$-1 \cdot 2^{2} \cdot 7^{2}$$ $$J_4$$ = $$1582$$ = $$2 \cdot 7 \cdot 113$$ $$J_6$$ = $$-17884$$ = $$-1 \cdot 2^{2} \cdot 17 \cdot 263$$ $$J_8$$ = $$250635$$ = $$3 \cdot 5 \cdot 7^{2} \cdot 11 \cdot 31$$ $$J_{10}$$ = $$-1088$$ = $$-1 \cdot 2^{6} \cdot 17$$ $$g_1$$ = $$4519603984/17$$ $$g_2$$ = $$186120718/17$$ $$g_3$$ = $$631463$$
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); $$\mathrm{Aut}(X_{\overline{\Q}})$$ $$\simeq$$ $$V_4$$ (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,-1,0],C![1,0,0]];

All rational points: (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: 1 (p = 2), 1 (p = 17) magma: TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);

Torsion: $$\Z/{6}\Z$$

### Sato-Tate group

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

### Decomposition

Splits over the number field $$\Q (b) \simeq$$ $$\Q(\sqrt{2})$$ with defining polynomial:
$$x^{2} - 2$$

Decomposes up to isogeny as the product of the non-isogenous elliptic curves:
Elliptic curve 2.2.8.1-17.2-a2
Elliptic curve 2.2.8.1-17.1-a1

### Endomorphisms

Not of $$\GL_2$$-type over $$\Q$$

Endomorphism ring over $$\Q$$:
 $$\End (J_{})$$ $$\simeq$$ $$\Z$$ $$\End (J_{}) \otimes \Q$$ $$\simeq$$ $$\Q$$ $$\End (J_{}) \otimes \R$$ $$\simeq$$ $$\R$$

Smallest field over which all endomorphisms are defined:
Galois number field $$K = \Q (a) \simeq$$ $$\Q(\sqrt{2})$$ with defining polynomial $$x^{2} - 2$$

Of $$\GL_2$$-type over $$\overline{\Q}$$

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