# Properties

 Label 100352.d.100352.1 Conductor 100352 Discriminant 100352 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, 4, 3, -4, -1, 1], R![]);
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-2, 4, 3, -4, -1, 1]), R([]))

$y^2 = x^5 - x^4 - 4x^3 + 3x^2 + 4x - 2$

## Invariants

 magma: Conductor(LSeries(C)); Factorization($1); $$N$$ = $$100352$$ = $$2^{11} \cdot 7^{2}$$ magma: Discriminant(C); Factorization(Integers()!$1); $$\Delta$$ = $$100352$$ = $$2^{11} \cdot 7^{2}$$

### G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$4864$$ = $$2^{8} \cdot 19$$ $$I_4$$ = $$351232$$ = $$2^{10} \cdot 7^{3}$$ $$I_6$$ = $$614154240$$ = $$2^{14} \cdot 3^{2} \cdot 5 \cdot 7^{2} \cdot 17$$ $$I_{10}$$ = $$411041792$$ = $$2^{23} \cdot 7^{2}$$ $$J_2$$ = $$608$$ = $$2^{5} \cdot 19$$ $$J_4$$ = $$11744$$ = $$2^{5} \cdot 367$$ $$J_6$$ = $$71936$$ = $$2^{8} \cdot 281$$ $$J_8$$ = $$-23546112$$ = $$-1 \cdot 2^{8} \cdot 3 \cdot 23 \cdot 31 \cdot 43$$ $$J_{10}$$ = $$100352$$ = $$2^{11} \cdot 7^{2}$$ $$g_1$$ = $$40568406016/49$$ $$g_2$$ = $$1288833536/49$$ $$g_3$$ = $$12984448/49$$
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,-1,1],C![1,0,0],C![1,1,1]];

All rational points: (1 : -1 : 1), (1 : 0 : 0), (1 : 1 : 1)

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

Number of rational Weierstrass points: $$1$$

## Invariants of the Jacobian:

Analytic rank: $$1$$

magma: TwoSelmerGroup(Jacobian(C)); NumberOfGenerators($1); 2-Selmer rank: $$2$$ magma: HasSquareSha(Jacobian(C)); Order of Ш*: square Tamagawa numbers: 1 (p = 2), 1 (p = 7) magma: TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);

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

### 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 $$\Q$$.