# Properties

 Label 1077.a.1077.1 Conductor 1077 Discriminant 1077 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

# Learn more about

Show commands for: Magma / SageMath

## Minimal equation

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![32, -90, -1, 80, 34, 5], R![1, 0, 0, 1]);
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([32, -90, -1, 80, 34, 5]), R([1, 0, 0, 1]))

$y^2 + (x^3 + 1)y = 5x^5 + 34x^4 + 80x^3 - x^2 - 90x + 32$

## Invariants

 magma: Conductor(LSeries(C)); Factorization($1); $$N$$ = $$1077$$ = $$3 \cdot 359$$ magma: Discriminant(C); Factorization(Integers()!$1); $$\Delta$$ = $$1077$$ = $$3 \cdot 359$$

### G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$311848$$ = $$2^{3} \cdot 17 \cdot 2293$$ $$I_4$$ = $$646372$$ = $$2^{2} \cdot 283 \cdot 571$$ $$I_6$$ = $$67039504232$$ = $$2^{3} \cdot 8379938029$$ $$I_{10}$$ = $$4411392$$ = $$2^{12} \cdot 3 \cdot 359$$ $$J_2$$ = $$38981$$ = $$17 \cdot 2293$$ $$J_4$$ = $$63306532$$ = $$2^{2} \cdot 61 \cdot 259453$$ $$J_6$$ = $$137068427976$$ = $$2^{3} \cdot 3 \cdot 5711184499$$ $$J_8$$ = $$333836849266358$$ = $$2 \cdot 20369 \cdot 8194728491$$ $$J_{10}$$ = $$1077$$ = $$3 \cdot 359$$ $$g_1$$ = $$90004636142290020118901/1077$$ $$g_2$$ = $$3749794358746968581012/1077$$ $$g_3$$ = $$69425997674312689112/359$$
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![-13,1094,2],C![-13,1095,2],C![1,-1,0],C![1,0,0]];

All rational points: (-13 : 1094 : 2), (-13 : 1095 : 2), (1 : -1 : 0), (1 : 0 : 0)

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

Number of rational Weierstrass points: $$0$$

## Invariants of the Jacobian:

Analytic rank: $$1$$

magma: TwoSelmerGroup(Jacobian(C)); NumberOfGenerators($1); 2-Selmer rank: $$2$$ magma: HasSquareSha(Jacobian(C)); Order of Ш*: square Tamagawa numbers: 1 (p = 3), 1 (p = 359) magma: TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);

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

### 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 $$\Q$$.