# Properties

 Label 1051.b.1051.2 Conductor 1051 Discriminant -1051 Sato-Tate group $\mathrm{USp}(4)$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\Q$$ $$\overline{\Q}$$-simple yes $$\mathrm{GL}_2$$-type no

# Related objects

Show commands for: Magma / SageMath

## Minimal equation

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 1, 0, 16, 8, 1], R![0, 1]);
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 1, 0, 16, 8, 1]), R([0, 1]))

$y^2 + xy = x^5 + 8x^4 + 16x^3 + x$

## Invariants

 magma: Conductor(LSeries(C)); Factorization($1); $$N$$ = $$1051$$ = $$1051$$ magma: Discriminant(C); Factorization(Integers()!$1); $$\Delta$$ = $$-1051$$ = $$-1 \cdot 1051$$

### G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$24704$$ = $$2^{7} \cdot 193$$ $$I_4$$ = $$-803840$$ = $$-1 \cdot 2^{10} \cdot 5 \cdot 157$$ $$I_6$$ = $$-6606414400$$ = $$-1 \cdot 2^{6} \cdot 5^{2} \cdot 67 \cdot 61627$$ $$I_{10}$$ = $$-4304896$$ = $$-1 \cdot 2^{12} \cdot 1051$$ $$J_2$$ = $$3088$$ = $$2^{4} \cdot 193$$ $$J_4$$ = $$405696$$ = $$2^{6} \cdot 3 \cdot 2113$$ $$J_6$$ = $$72449921$$ = $$72449921$$ $$J_8$$ = $$14784027908$$ = $$2^{2} \cdot 13 \cdot 19 \cdot 313 \cdot 47807$$ $$J_{10}$$ = $$-1051$$ = $$-1 \cdot 1051$$ $$g_1$$ = $$-280793117300359168/1051$$ $$g_2$$ = $$-11946277554880512/1051$$ $$g_3$$ = $$-690863899476224/1051$$
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); $$\mathrm{Aut}(X_{\overline{\Q}})$$ $$\simeq$$ $$C_2$$ (GAP id : [2,1])

## 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![-4,2,1],C![0,0,1],C![1,0,0]];

All rational points: (-4 : 2 : 1), (0 : 0 : 1), (1 : 0 : 0)

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

Number of rational Weierstrass points: $$3$$

## 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 = 1051) magma: TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);

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

### Sato-Tate group

 $$\mathrm{ST}$$ $$\simeq$$ $\mathrm{USp}(4)$ $$\mathrm{ST}^0$$ $$\simeq$$ $$\mathrm{USp}(4)$$

### Decomposition

Simple over $$\overline{\Q}$$

### Endomorphisms

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

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

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