Properties

 Label 3812.a.243968.1 Conductor 3812 Discriminant 243968 Mordell-Weil group $$\Z$$ 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: SageMath / Magma

Simplified equation

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

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

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

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

magma: X,pi:= SimplifiedModel(C);

Invariants

 Conductor: $$N$$ $$=$$ $$3812$$ $$=$$ $$2^{2} \cdot 953$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$243968$$ $$=$$ $$2^{8} \cdot 953$$ magma: Discriminant(C); Factorization(Integers()!$1);

G2 invariants

 $$I_2$$ $$=$$ $$-216$$ $$=$$ $$- 2^{3} \cdot 3^{3}$$ $$I_4$$ $$=$$ $$50532$$ $$=$$ $$2^{2} \cdot 3 \cdot 4211$$ $$I_6$$ $$=$$ $$-6364440$$ $$=$$ $$- 2^{3} \cdot 3^{3} \cdot 5 \cdot 71 \cdot 83$$ $$I_{10}$$ $$=$$ $$999292928$$ $$=$$ $$2^{20} \cdot 953$$ $$J_2$$ $$=$$ $$-27$$ $$=$$ $$- 3^{3}$$ $$J_4$$ $$=$$ $$-496$$ $$=$$ $$- 2^{4} \cdot 31$$ $$J_6$$ $$=$$ $$7056$$ $$=$$ $$2^{4} \cdot 3^{2} \cdot 7^{2}$$ $$J_8$$ $$=$$ $$-109132$$ $$=$$ $$- 2^{2} \cdot 27283$$ $$J_{10}$$ $$=$$ $$243968$$ $$=$$ $$2^{8} \cdot 953$$ $$g_1$$ $$=$$ $$-14348907/243968$$ $$g_2$$ $$=$$ $$610173/15248$$ $$g_3$$ $$=$$ $$321489/15248$$

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
$$(1 : 0 : 0)$$ $$(1 : -1 : 0)$$ $$(0 : 0 : 1)$$ $$(0 : -1 : 1)$$ $$(1 : 0 : 1)$$ $$(-1 : -2 : 1)$$
$$(-1 : 2 : 1)$$ $$(1 : -2 : 1)$$ $$(5 : 10 : 3)$$ $$(5 : -162 : 3)$$

magma: [C![-1,-2,1],C![-1,2,1],C![0,-1,1],C![0,0,1],C![1,-2,1],C![1,-1,0],C![1,0,0],C![1,0,1],C![5,-162,3],C![5,10,3]];

Number of rational Weierstrass points: $$0$$

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: $$\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(1 : -2 : 1) - (1 : 0 : 0)$$ $$z (x - z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-x^3 - z^3$$ $$0.002640$$ $$\infty$$

BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$1$$ Mordell-Weil rank: $$1$$ 2-Selmer rank: $$1$$ Regulator: $$0.002640$$ Real period: $$14.80210$$ Tamagawa product: $$13$$ Torsion order: $$1$$ Leading coefficient: $$0.508099$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$2$$ $$8$$ $$13$$ $$( 1 - T )( 1 + T )$$
$$953$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 + 22 T + 953 T^{2} )$$

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 $$\Q$$.