# Properties

 Label 100010.a.400040.1 Conductor 100010 Discriminant -400040 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![5, 2, -10, 1, 2, 1, -1], R![1, 0, 0, 1]);
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([5, 2, -10, 1, 2, 1, -1]), R([1, 0, 0, 1]))

$y^2 + (x^3 + 1)y = -x^6 + x^5 + 2x^4 + x^3 - 10x^2 + 2x + 5$

## Invariants

 magma: Conductor(LSeries(C)); Factorization($1); $$N$$ = $$100010$$ = $$2 \cdot 5 \cdot 73 \cdot 137$$ magma: Discriminant(C); Factorization(Integers()!$1); $$\Delta$$ = $$-400040$$ = $$-1 \cdot 2^{3} \cdot 5 \cdot 73 \cdot 137$$

### G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$21736$$ = $$2^{3} \cdot 11 \cdot 13 \cdot 19$$ $$I_4$$ = $$-69788$$ = $$-1 \cdot 2^{2} \cdot 73 \cdot 239$$ $$I_6$$ = $$-552591640$$ = $$-1 \cdot 2^{3} \cdot 5 \cdot 59 \cdot 234149$$ $$I_{10}$$ = $$-1638563840$$ = $$-1 \cdot 2^{15} \cdot 5 \cdot 73 \cdot 137$$ $$J_2$$ = $$2717$$ = $$11 \cdot 13 \cdot 19$$ $$J_4$$ = $$308314$$ = $$2 \cdot 154157$$ $$J_6$$ = $$46839264$$ = $$2^{5} \cdot 3 \cdot 31 \cdot 15739$$ $$J_8$$ = $$8051189423$$ = $$4219 \cdot 1908317$$ $$J_{10}$$ = $$-400040$$ = $$-1 \cdot 2^{3} \cdot 5 \cdot 73 \cdot 137$$ $$g_1$$ = $$-148063561656653357/400040$$ $$g_2$$ = $$-3091947885524641/200020$$ $$g_3$$ = $$-43221451942812/50005$$
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,-2,1],C![1,0,1],C![21,-5792,13],C![21,-5666,13]];

All rational points: (1 : -2 : 1), (1 : 0 : 1), (21 : -5792 : 13), (21 : -5666 : 13)

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

Number of rational Weierstrass points: $$0$$

## 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: 3 (p = 2), 1 (p = 5), 1 (p = 73), 1 (p = 137) magma: TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);

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

2-torsion field: splitting field of $$x^{6} - 3 x^{5} - 50 x^{4} + 105 x^{3} + 1525 x^{2} - 1578 x - 19526$$ with Galois group $S_4\times C_2$

### 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 $$\Q$$.