# Properties

 Label 3571.a.3571.1 Conductor 3571 Discriminant -3571 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, 1, 0, -2], R![1, 1, 0, 1]);

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

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

## Invariants

 magma: Conductor(LSeries(C)); Factorization($1); $$N$$ = $$3571$$ = $$3571$$ magma: Discriminant(C); Factorization(Integers()!$1); $$\Delta$$ = $$-3571$$ = $$-1 \cdot 3571$$

### G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$264$$ = $$2^{3} \cdot 3 \cdot 11$$ $$I_4$$ = $$15396$$ = $$2^{2} \cdot 3 \cdot 1283$$ $$I_6$$ = $$246696$$ = $$2^{3} \cdot 3 \cdot 19 \cdot 541$$ $$I_{10}$$ = $$-14626816$$ = $$-1 \cdot 2^{12} \cdot 3571$$ $$J_2$$ = $$33$$ = $$3 \cdot 11$$ $$J_4$$ = $$-115$$ = $$-1 \cdot 5 \cdot 23$$ $$J_6$$ = $$1125$$ = $$3^{2} \cdot 5^{3}$$ $$J_8$$ = $$5975$$ = $$5^{2} \cdot 239$$ $$J_{10}$$ = $$-3571$$ = $$-1 \cdot 3571$$ $$g_1$$ = $$-39135393/3571$$ $$g_2$$ = $$4132755/3571$$ $$g_3$$ = $$-1225125/3571$$
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,-8,2],C![-1,0,1],C![-1,1,1],C![-1,5,2],C![0,-1,1],C![0,0,1],C![1,-2,1],C![1,-1,0],C![1,-1,1],C![1,0,0],C![2,-6,1],C![2,-5,1],C![20,-18045,21],C![20,-8036,21]];

Known rational points: (-1 : -8 : 2), (-1 : 0 : 1), (-1 : 1 : 1), (-1 : 5 : 2), (0 : -1 : 1), (0 : 0 : 1), (1 : -2 : 1), (1 : -1 : 0), (1 : -1 : 1), (1 : 0 : 0), (2 : -6 : 1), (2 : -5 : 1), (20 : -18045 : 21), (20 : -8036 : 21)

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: $$2$$ magma: HasSquareSha(Jacobian(C)); Order of Ш*: square Regulator: 0.00753020367043 Real period: 26.618204999994126386695233697 Tamagawa numbers: 1 (p = 3571) magma: TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);

Torsion: $$\mathrm{trivial}$$

### 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 $$\Q$$.