# Properties

 Label 100224.b.601344.1 Conductor 100224 Discriminant 601344 Mordell-Weil group $$\Z \times \Z/{2}\Z$$ 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

## Simplified equation

 $y^2 = x^5 + x^4 - 2x^3 + 4x^2 - 2x + 1$ (homogenize, simplify) $y^2 = x^5z + x^4z^2 - 2x^3z^3 + 4x^2z^4 - 2xz^5 + z^6$ (dehomogenize, simplify) $y^2 = x^5 + x^4 - 2x^3 + 4x^2 - 2x + 1$ (minimize, homogenize)

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![1, -2, 4, -2, 1, 1], R![]);

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([1, -2, 4, -2, 1, 1]), R([]));

magma: X,pi:= SimplifiedModel(C);

sage: X = HyperellipticCurve(R([1, -2, 4, -2, 1, 1]))

## Invariants

 Conductor: $$N$$ = $$100224$$ = $$2^{7} \cdot 3^{3} \cdot 29$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ = $$601344$$ = $$2^{8} \cdot 3^{4} \cdot 29$$ magma: Discriminant(C); Factorization(Integers()!$1);

### G2 invariants

 $$I_2$$ = $$-1920$$ = $$- 2^{7} \cdot 3 \cdot 5$$ $$I_4$$ = $$350208$$ = $$2^{11} \cdot 3^{2} \cdot 19$$ $$I_6$$ = $$-157777920$$ = $$- 2^{15} \cdot 3^{2} \cdot 5 \cdot 107$$ $$I_{10}$$ = $$2463105024$$ = $$2^{20} \cdot 3^{4} \cdot 29$$ $$J_2$$ = $$-240$$ = $$- 2^{4} \cdot 3 \cdot 5$$ $$J_4$$ = $$-1248$$ = $$- 2^{5} \cdot 3 \cdot 13$$ $$J_6$$ = $$-1280$$ = $$- 2^{8} \cdot 5$$ $$J_8$$ = $$-312576$$ = $$- 2^{8} \cdot 3 \cdot 11 \cdot 37$$ $$J_{10}$$ = $$601344$$ = $$2^{8} \cdot 3^{4} \cdot 29$$ $$g_1$$ = $$-38400000/29$$ $$g_2$$ = $$832000/29$$ $$g_3$$ = $$-32000/261$$

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$magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);

## Rational points

All points: $$(1 : 0 : 0),\, (0 : -1 : 1),\, (0 : 1 : 1),\, (-1 : -3 : 1),\, (-1 : 3 : 1)$$

magma: [C![-1,-3,1],C![-1,3,1],C![0,-1,1],C![0,1,1],C![1,0,0]];

Number of rational Weierstrass points: $$1$$

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

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(-1 : -3 : 1) + (0 : -1 : 1) - 2 \cdot(1 : 0 : 0)$$ $$x (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$2xz^2 - z^3$$ $$0.191300$$ $$\infty$$
$$D_0 - 2 \cdot(1 : 0 : 0)$$ $$x^2 - xz + z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0$$ $$2$$

## BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$1$$ Mordell-Weil rank: $$1$$ 2-Selmer rank: $$2$$ Regulator: $$0.191300$$ Real period: $$10.21193$$ Tamagawa product: $$4$$ Torsion order: $$2$$ Leading coefficient: $$1.953549$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

## Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$2$$ $$8$$ $$7$$ $$2$$ $$1$$
$$3$$ $$4$$ $$3$$ $$2$$ $$1 + T$$
$$29$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 + 4 T + 29 T^{2} )$$

## Sato-Tate group

 $$\mathrm{ST}$$ $$\simeq$$ $\mathrm{USp}(4)$ $$\mathrm{ST}^0$$ $$\simeq$$ $$\mathrm{USp}(4)$$

## Decomposition of the Jacobian

Simple over $$\overline{\Q}$$

## Endomorphisms of the Jacobian

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

Endomorphism ring over $$\Q$$:

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

All $$\overline{\Q}$$-endomorphisms of the Jacobian are defined over $$\Q$$.