# Properties

 Label 100352.a.100352.1 Conductor 100352 Discriminant 100352 Mordell-Weil group $$\Z/{2}\Z$$ 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

## Simplified equation

 $y^2 = x^5 - x^4 + x^2 + x$ (homogenize, simplify) $y^2 = x^5z - x^4z^2 + x^2z^4 + xz^5$ (dehomogenize, simplify) $y^2 = x^5 - x^4 + x^2 + x$ (minimize, homogenize)

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([]));

magma: X,pi:= SimplifiedModel(C);

sage: X = HyperellipticCurve(R([0, 1, 1, 0, -1, 1]))

## Invariants

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

### G2 invariants

 $$I_2$$ = $$896$$ = $$2^{7} \cdot 7$$ $$I_4$$ = $$-20480$$ = $$- 2^{12} \cdot 5$$ $$I_6$$ = $$-1654784$$ = $$- 2^{14} \cdot 101$$ $$I_{10}$$ = $$411041792$$ = $$2^{23} \cdot 7^{2}$$ $$J_2$$ = $$112$$ = $$2^{4} \cdot 7$$ $$J_4$$ = $$736$$ = $$2^{5} \cdot 23$$ $$J_6$$ = $$-512$$ = $$- 2^{9}$$ $$J_8$$ = $$-149760$$ = $$- 2^{8} \cdot 3^{2} \cdot 5 \cdot 13$$ $$J_{10}$$ = $$100352$$ = $$2^{11} \cdot 7^{2}$$ $$g_1$$ = $$175616$$ $$g_2$$ = $$10304$$ $$g_3$$ = $$-64$$

magma: IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);

sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]

## Automorphism group

 $$\mathrm{Aut}(X)$$ $$\simeq$$ $C_2$ magma: AutomorphismGroup(C); IdentifyGroup($1); $$\mathrm{Aut}(X_{\overline{\Q}})$$ $$\simeq$$$C_2^2$magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);

## Rational points

All points: $$(1 : 0 : 0),\, (0 : 0 : 1)$$

magma: [C![0,0,1],C![1,0,0]];

Number of rational Weierstrass points: $$2$$

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

This curve is locally solvable everywhere.

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);

## Mordell-Weil group of the Jacobian:

Group structure: $$\Z/{2}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(0 : 0 : 1) - (1 : 0 : 0)$$ $$x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0$$ $$2$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$1$$ Regulator: $$1$$ Real period: $$8.501977$$ Tamagawa product: $$1$$ Torsion order: $$2$$ Leading coefficient: $$2.125494$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$2$$ $$11$$ $$11$$ $$1$$ $$1$$
$$7$$ $$2$$ $$2$$ $$1$$ $$1 + T^{2}$$

## Sato-Tate group

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

## Decomposition of the Jacobian

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 of the Jacobian

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 $$\Z \times \Z$$ $$\End (J_{\overline{\Q}}) \otimes \Q$$ $$\simeq$$ $$\Q$$ $$\times$$ $$\Q$$ $$\End (J_{\overline{\Q}}) \otimes \R$$ $$\simeq$$ $$\R \times \R$$