# Properties

 Label 100352.c.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![-14, 14, 7, -9, 0, 1], R![]);
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-14, 14, 7, -9, 0, 1]), R([]))

$y^2 = x^5 - 9x^3 + 7x^2 + 14x - 14$

## 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);

 $$I_2$$ = $$16736$$ = $$2^{5} \cdot 523$$ $$I_4$$ = $$265216$$ = $$2^{10} \cdot 7 \cdot 37$$ $$I_6$$ = $$1458946048$$ = $$2^{14} \cdot 7 \cdot 12721$$ $$I_{10}$$ = $$411041792$$ = $$2^{23} \cdot 7^{2}$$ $$J_2$$ = $$2092$$ = $$2^{2} \cdot 523$$ $$J_4$$ = $$179590$$ = $$2 \cdot 5 \cdot 17959$$ $$J_6$$ = $$20265956$$ = $$2^{2} \cdot 71 \cdot 71359$$ $$J_8$$ = $$2535952963$$ = $$2535952963$$ $$J_{10}$$ = $$100352$$ = $$2^{11} \cdot 7^{2}$$ $$g_1$$ = $$39129873538843/98$$ $$g_2$$ = $$12845683618265/784$$ $$g_3$$ = $$1385831669681/1568$$
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],C![9,-182,4],C![9,182,4]];

All rational points: (1 : 0 : 0), (9 : -182 : 4), (9 : 182 : 4)

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 $$\Q$$.