# Properties

 Label 5705.a.5705.1 Conductor 5705 Discriminant -5705 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, 2, 4, 1, 1, 1], R![1, 0, 0, 1]);

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 2, 4, 1, 1, 1]), R([1, 0, 0, 1]))

$y^2 + (x^3 + 1)y = x^5 + x^4 + x^3 + 4x^2 + 2x$

## Invariants

 magma: Conductor(LSeries(C)); Factorization($1); $$N$$ = $$5705$$ = $$5 \cdot 7 \cdot 163$$ magma: Discriminant(C); Factorization(Integers()!$1); $$\Delta$$ = $$-5705$$ = $$-1 \cdot 5 \cdot 7 \cdot 163$$

### G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$232$$ = $$2^{3} \cdot 29$$ $$I_4$$ = $$62692$$ = $$2^{2} \cdot 7 \cdot 2239$$ $$I_6$$ = $$-1404824$$ = $$-1 \cdot 2^{3} \cdot 41 \cdot 4283$$ $$I_{10}$$ = $$-23367680$$ = $$-1 \cdot 2^{12} \cdot 5 \cdot 7 \cdot 163$$ $$J_2$$ = $$29$$ = $$29$$ $$J_4$$ = $$-618$$ = $$-1 \cdot 2 \cdot 3 \cdot 103$$ $$J_6$$ = $$7756$$ = $$2^{2} \cdot 7 \cdot 277$$ $$J_8$$ = $$-39250$$ = $$-1 \cdot 2 \cdot 5^{3} \cdot 157$$ $$J_{10}$$ = $$-5705$$ = $$-1 \cdot 5 \cdot 7 \cdot 163$$ $$g_1$$ = $$-20511149/5705$$ $$g_2$$ = $$15072402/5705$$ $$g_3$$ = $$-931828/815$$
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![-24,4304,7],C![-24,9177,7],C![-2,3,1],C![-2,4,1],C![-1,-6,2],C![-1,-1,1],C![-1,-1,2],C![-1,1,1],C![0,-1,1],C![0,0,1],C![1,-1,0],C![1,0,0]];

Known rational points: (-24 : 4304 : 7), (-24 : 9177 : 7), (-2 : 3 : 1), (-2 : 4 : 1), (-1 : -6 : 2), (-1 : -1 : 1), (-1 : -1 : 2), (-1 : 1 : 1), (0 : -1 : 1), (0 : 0 : 1), (1 : -1 : 0), (1 : 0 : 0)

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

Number of rational Weierstrass points: $$0$$

## Invariants of the Jacobian:

Analytic rank*: $$2$$

magma: TwoSelmerGroup(Jacobian(C)); NumberOfGenerators($1); 2-Selmer rank: $$3$$ magma: HasSquareSha(Jacobian(C)); Order of Ш*: square Regulator: 0.0444276518743 Real period: 27.322834256046468421774959788 Tamagawa numbers: 1 (p = 5), 1 (p = 7), 1 (p = 163) magma: TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);

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

2-torsion field: splitting field of $$x^{6} - x^{5} - x^{4} - 28 x^{3} + 94 x^{2} + 440 x + 296$$ with Galois group $S_4\times C_2$

### 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 $$\Q$$.