# Properties

 Label 100036.a.200072.1 Conductor 100036 Discriminant 200072 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![-6, 12, -1, -8, 0, 1], R![1, 0, 1, 1]);
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-6, 12, -1, -8, 0, 1]), R([1, 0, 1, 1]))

$y^2 + (x^3 + x^2 + 1)y = x^5 - 8x^3 - x^2 + 12x - 6$

## Invariants

 magma: Conductor(LSeries(C)); Factorization($1); $$N$$ = $$100036$$ = $$2^{2} \cdot 89 \cdot 281$$ magma: Discriminant(C); Factorization(Integers()!$1); $$\Delta$$ = $$200072$$ = $$2^{3} \cdot 89 \cdot 281$$

### G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$22472$$ = $$2^{3} \cdot 53^{2}$$ $$I_4$$ = $$346180$$ = $$2^{2} \cdot 5 \cdot 19 \cdot 911$$ $$I_6$$ = $$2597903816$$ = $$2^{3} \cdot 1019 \cdot 318683$$ $$I_{10}$$ = $$819494912$$ = $$2^{15} \cdot 89 \cdot 281$$ $$J_2$$ = $$2809$$ = $$53^{2}$$ $$J_4$$ = $$325164$$ = $$2^{2} \cdot 3 \cdot 7^{3} \cdot 79$$ $$J_6$$ = $$49609856$$ = $$2^{7} \cdot 387577$$ $$J_8$$ = $$8405614652$$ = $$2^{2} \cdot 2101403663$$ $$J_{10}$$ = $$200072$$ = $$2^{3} \cdot 89 \cdot 281$$ $$g_1$$ = $$174887470365513049/200072$$ $$g_2$$ = $$1801763080537539/50018$$ $$g_3$$ = $$48930703272592/25009$$
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![-2,1,1],C![-2,2,1],C![1,-2,1],C![1,-1,0],C![1,-1,1],C![1,0,0],C![4,-78,3],C![4,-61,3]];

Known rational points: (-2 : 1 : 1), (-2 : 2 : 1), (1 : -2 : 1), (1 : -1 : 0), (1 : -1 : 1), (1 : 0 : 0), (4 : -78 : 3), (4 : -61 : 3)

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 Tamagawa numbers: 3 (p = 2), 1 (p = 89), 1 (p = 281) 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 $$\Q$$.