# Properties

 Label 1104.a.17664.1 Conductor 1104 Discriminant 17664 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![-1, 3, -4, 4, -2, 1], R![]);
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-1, 3, -4, 4, -2, 1]), R([]))

$y^2 = x^5 - 2x^4 + 4x^3 - 4x^2 + 3x - 1$

## Invariants

 magma: Conductor(LSeries(C)); Factorization($1); $$N$$ = $$1104$$ = $$2^{4} \cdot 3 \cdot 23$$ magma: Discriminant(C); Factorization(Integers()!$1); $$\Delta$$ = $$17664$$ = $$2^{8} \cdot 3 \cdot 23$$

### G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$1408$$ = $$2^{7} \cdot 11$$ $$I_4$$ = $$40960$$ = $$2^{13} \cdot 5$$ $$I_6$$ = $$20021248$$ = $$2^{15} \cdot 13 \cdot 47$$ $$I_{10}$$ = $$72351744$$ = $$2^{20} \cdot 3 \cdot 23$$ $$J_2$$ = $$176$$ = $$2^{4} \cdot 11$$ $$J_4$$ = $$864$$ = $$2^{5} \cdot 3^{3}$$ $$J_6$$ = $$-1280$$ = $$-1 \cdot 2^{8} \cdot 5$$ $$J_8$$ = $$-242944$$ = $$-1 \cdot 2^{8} \cdot 13 \cdot 73$$ $$J_{10}$$ = $$17664$$ = $$2^{8} \cdot 3 \cdot 23$$ $$g_1$$ = $$659664896/69$$ $$g_2$$ = $$6133248/23$$ $$g_3$$ = $$-154880/69$$
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![1,-1,1],C![1,0,0],C![1,1,1]];

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

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

Number of rational Weierstrass points: $$1$$

## 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: 5 (p = 2), 1 (p = 3), 1 (p = 23) magma: TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);

Torsion: $$\Z/{10}\Z$$

### 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 $$\Q$$.