# Properties

 Label 100352.a.100352.1 Conductor 100352 Discriminant 100352 Sato-Tate group $N(G_{3,3})$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R \times \R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\Q \times \Q$$ $$\overline{\Q}$$-simple no $$\mathrm{GL}_2$$-type no

# Related objects

Show commands for: Magma / SageMath

## Minimal equation

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 1, 1, 0, -1, 1], R![]);
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 1, 1, 0, -1, 1]), R([]))

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

## 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);

 $$I_2$$ = $$896$$ = $$2^{7} \cdot 7$$ $$I_4$$ = $$-20480$$ = $$-1 \cdot 2^{12} \cdot 5$$ $$I_6$$ = $$-1654784$$ = $$-1 \cdot 2^{14} \cdot 101$$ $$I_{10}$$ = $$411041792$$ = $$2^{23} \cdot 7^{2}$$ $$J_2$$ = $$112$$ = $$2^{4} \cdot 7$$ $$J_4$$ = $$736$$ = $$2^{5} \cdot 23$$ $$J_6$$ = $$-512$$ = $$-1 \cdot 2^{9}$$ $$J_8$$ = $$-149760$$ = $$-1 \cdot 2^{8} \cdot 3^{2} \cdot 5 \cdot 13$$ $$J_{10}$$ = $$100352$$ = $$2^{11} \cdot 7^{2}$$ $$g_1$$ = $$175616$$ $$g_2$$ = $$10304$$ $$g_3$$ = $$-64$$
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); $$\mathrm{Aut}(X_{\overline{\Q}})$$ $$\simeq$$ $$V_4$$ (GAP id : [4,2])

## 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![0,0,1],C![1,0,0]];

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

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

Number of rational Weierstrass points: $$2$$

## Invariants of the Jacobian:

Analytic rank: $$0$$

magma: TwoSelmerGroup(Jacobian(C)); NumberOfGenerators($1); 2-Selmer rank: $$1$$ 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

 $$\mathrm{ST}$$ $$\simeq$$ $N(G_{3,3})$ $$\mathrm{ST}^0$$ $$\simeq$$ $$\mathrm{SU}(2)\times\mathrm{SU}(2)$$

### Decomposition

Splits over the number field $$\Q (b) \simeq$$ $$\Q(\sqrt{-1})$$ with defining polynomial:
$$x^{2} + 1$$

Decomposes up to isogeny as the product of the non-isogenous elliptic curves:
Elliptic curve 2.0.4.1-6272.1-b1
Elliptic curve 2.0.4.1-6272.1-a1

### Endomorphisms

Not of $$\GL_2$$-type over $$\Q$$

Endomorphism ring over $$\Q$$:
 $$\End (J_{})$$ $$\simeq$$ $$\Z$$ $$\End (J_{}) \otimes \Q$$ $$\simeq$$ $$\Q$$ $$\End (J_{}) \otimes \R$$ $$\simeq$$ $$\R$$

Smallest field over which all endomorphisms are defined:
Galois number field $$K = \Q (a) \simeq$$ $$\Q(\sqrt{-1})$$ with defining polynomial $$x^{2} + 1$$

Of $$\GL_2$$-type over $$\overline{\Q}$$

Endomorphism ring over $$\overline{\Q}$$:
 $$\End (J_{\overline{\Q}})$$ $$\simeq$$ an order of index $$2$$ in $$\Z \times \Z$$ $$\End (J_{\overline{\Q}}) \otimes \Q$$ $$\simeq$$ $$\Q$$ $$\times$$ $$\Q$$ $$\End (J_{\overline{\Q}}) \otimes \R$$ $$\simeq$$ $$\R \times \R$$