# Properties

 Label 100128.a.801024.1 Conductor 100128 Discriminant 801024 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![-2, 7, -2, -13, -1, 4], R![1, 0, 1]);
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-2, 7, -2, -13, -1, 4]), R([1, 0, 1]))

$y^2 + (x^2 + 1)y = 4x^5 - x^4 - 13x^3 - 2x^2 + 7x - 2$

## Invariants

 magma: Conductor(LSeries(C)); Factorization($1); $$N$$ = $$100128$$ = $$2^{5} \cdot 3 \cdot 7 \cdot 149$$ magma: Discriminant(C); Factorization(Integers()!$1); $$\Delta$$ = $$801024$$ = $$2^{8} \cdot 3 \cdot 7 \cdot 149$$

### G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$33856$$ = $$2^{6} \cdot 23^{2}$$ $$I_4$$ = $$34483648$$ = $$2^{6} \cdot 269 \cdot 2003$$ $$I_6$$ = $$317292855552$$ = $$2^{8} \cdot 3^{2} \cdot 137713913$$ $$I_{10}$$ = $$3280994304$$ = $$2^{20} \cdot 3 \cdot 7 \cdot 149$$ $$J_2$$ = $$4232$$ = $$2^{3} \cdot 23^{2}$$ $$J_4$$ = $$387038$$ = $$2 \cdot 431 \cdot 449$$ $$J_6$$ = $$46859332$$ = $$2^{2} \cdot 13 \cdot 901141$$ $$J_8$$ = $$12127569895$$ = $$5 \cdot 7 \cdot 307 \cdot 397 \cdot 2843$$ $$J_{10}$$ = $$801024$$ = $$2^{8} \cdot 3 \cdot 7 \cdot 149$$ $$g_1$$ = $$5302593435347072/3129$$ $$g_2$$ = $$114591028813564/3129$$ $$g_3$$ = $$3278290581553/3129$$
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![1,0,0]];

All rational points: (1 : 0 : 0)

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: $$0$$ magma: HasSquareSha(Jacobian(C)); Order of Ш*: square Tamagawa numbers: 2 (p = 2), 1 (p = 3), 1 (p = 7), 1 (p = 149) 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 $$\Q$$.