# Properties

 Label 11944.a.95552.1 Conductor 11944 Discriminant -95552 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, 1, -1, -2, 0, 1], R![0, 1, 0, 1]);

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

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

## Invariants

 magma: Conductor(LSeries(C)); Factorization($1); $$N$$ = $$11944$$ = $$2^{3} \cdot 1493$$ magma: Discriminant(C); Factorization(Integers()!$1); $$\Delta$$ = $$-95552$$ = $$-1 \cdot 2^{6} \cdot 1493$$

### G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$160$$ = $$2^{5} \cdot 5$$ $$I_4$$ = $$88576$$ = $$2^{9} \cdot 173$$ $$I_6$$ = $$963328$$ = $$2^{8} \cdot 53 \cdot 71$$ $$I_{10}$$ = $$-391380992$$ = $$-1 \cdot 2^{18} \cdot 1493$$ $$J_2$$ = $$20$$ = $$2^{2} \cdot 5$$ $$J_4$$ = $$-906$$ = $$-1 \cdot 2 \cdot 3 \cdot 151$$ $$J_6$$ = $$3472$$ = $$2^{4} \cdot 7 \cdot 31$$ $$J_8$$ = $$-187849$$ = $$-1 \cdot 37 \cdot 5077$$ $$J_{10}$$ = $$-95552$$ = $$-1 \cdot 2^{6} \cdot 1493$$ $$g_1$$ = $$-50000/1493$$ $$g_2$$ = $$113250/1493$$ $$g_3$$ = $$-21700/1493$$
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![-3,10,1],C![-3,20,1],C![-2,3,1],C![-2,7,1],C![-1,0,1],C![-1,2,1],C![0,-1,1],C![0,1,1],C![1,-11,2],C![1,-2,1],C![1,-1,0],C![1,0,0],C![1,0,1],C![1,6,2]];

Known rational points: (-3 : 10 : 1), (-3 : 20 : 1), (-2 : 3 : 1), (-2 : 7 : 1), (-1 : 0 : 1), (-1 : 2 : 1), (0 : -1 : 1), (0 : 1 : 1), (1 : -11 : 2), (1 : -2 : 1), (1 : -1 : 0), (1 : 0 : 0), (1 : 0 : 1), (1 : 6 : 2)

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: 0.00465674830447 Real period: 19.235030079544825801694536562 Tamagawa numbers: 6 (p = 2), 1 (p = 1493) 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 $$\Q$$.