Properties

 Label 1180.a.18880.1 Conductor $1180$ Discriminant $-18880$ Mordell-Weil group $$\Z/{18}\Z$$ Sato-Tate group $\mathrm{USp}(4)$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\Q$$ $$\End(J) \otimes \Q$$ $$\Q$$ $$\overline{\Q}$$-simple yes $$\mathrm{GL}_2$$-type no

Related objects

Show commands: SageMath / Magma

Simplified equation

 $y^2 + (x^3 + 1)y = -2x^4 + 4x^2 + 2x$ (homogenize, simplify) $y^2 + (x^3 + z^3)y = -2x^4z^2 + 4x^2z^4 + 2xz^5$ (dehomogenize, simplify) $y^2 = x^6 - 8x^4 + 2x^3 + 16x^2 + 8x + 1$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 2, 4, 0, -2]), R([1, 0, 0, 1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 2, 4, 0, -2], R![1, 0, 0, 1]);

sage: X = HyperellipticCurve(R([1, 8, 16, 2, -8, 0, 1]))

magma: X,pi:= SimplifiedModel(C);

Invariants

 Conductor: $$N$$ $$=$$ $$1180$$ $$=$$ $$2^{2} \cdot 5 \cdot 59$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$-18880$$ $$=$$ $$- 2^{6} \cdot 5 \cdot 59$$ magma: Discriminant(C); Factorization(Integers()!$1);

G2 invariants

 $$I_2$$ $$=$$ $$916$$ $$=$$ $$2^{2} \cdot 229$$ $$I_4$$ $$=$$ $$23257$$ $$=$$ $$13 \cdot 1789$$ $$I_6$$ $$=$$ $$5960477$$ $$=$$ $$359 \cdot 16603$$ $$I_{10}$$ $$=$$ $$-2416640$$ $$=$$ $$- 2^{13} \cdot 5 \cdot 59$$ $$J_2$$ $$=$$ $$229$$ $$=$$ $$229$$ $$J_4$$ $$=$$ $$1216$$ $$=$$ $$2^{6} \cdot 19$$ $$J_6$$ $$=$$ $$6656$$ $$=$$ $$2^{9} \cdot 13$$ $$J_8$$ $$=$$ $$11392$$ $$=$$ $$2^{7} \cdot 89$$ $$J_{10}$$ $$=$$ $$-18880$$ $$=$$ $$- 2^{6} \cdot 5 \cdot 59$$ $$g_1$$ $$=$$ $$-629763392149/18880$$ $$g_2$$ $$=$$ $$-228170791/295$$ $$g_3$$ $$=$$ $$-5453864/295$$

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

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

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

magma: [C![-1,0,1],C![0,-1,1],C![0,0,1],C![1,-1,0],C![1,0,0]]; // minimal model

magma: [C![-1,0,1],C![0,-1,1],C![0,1,1],C![1,-1,0],C![1,1,0]]; // simplified model

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: $$\Z/{18}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(-1 : 0 : 1) - (1 : 0 : 0)$$ $$z (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-x^3 - z^3$$ $$0$$ $$18$$
Generator $D_0$ Height Order
$$(-1 : 0 : 1) - (1 : 0 : 0)$$ $$z (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-x^3 - z^3$$ $$0$$ $$18$$
Generator $D_0$ Height Order
$$(-1 : 0 : 1) - (1 : 1 : 0)$$ $$z (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-x^3 - z^3$$ $$0$$ $$18$$

BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$1$$ Regulator: $$1$$ Real period: $$24.17051$$ Tamagawa product: $$6$$ Torsion order: $$18$$ Leading coefficient: $$0.447602$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$2$$ $$6$$ $$6$$ $$( 1 - T )( 1 + T )$$
$$5$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + 3 T + 5 T^{2} )$$
$$59$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 + 12 T + 59 T^{2} )$$

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 $$\Q$$.