Properties

 Label 1062.a.6372.1 Conductor $1062$ Discriminant $6372$ Mordell-Weil group $$\Z \times \Z/{2}\Z$$ Sato-Tate group $\mathrm{USp}(4)$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\Q$$ $$\End(J) \otimes \Q$$ $$\Q$$ $$\overline{\Q}$$-simple yes $$\mathrm{GL}_2$$-type no

Related objects

Show commands: SageMath / Magma

Simplified equation

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

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

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

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

magma: X,pi:= SimplifiedModel(C);

Invariants

 Conductor: $$N$$ $$=$$ $$1062$$ $$=$$ $$2 \cdot 3^{2} \cdot 59$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$6372$$ $$=$$ $$2^{2} \cdot 3^{3} \cdot 59$$ magma: Discriminant(C); Factorization(Integers()!$1);

G2 invariants

 $$I_2$$ $$=$$ $$300$$ $$=$$ $$2^{2} \cdot 3 \cdot 5^{2}$$ $$I_4$$ $$=$$ $$2601$$ $$=$$ $$3^{2} \cdot 17^{2}$$ $$I_6$$ $$=$$ $$306603$$ $$=$$ $$3^{2} \cdot 11 \cdot 19 \cdot 163$$ $$I_{10}$$ $$=$$ $$-815616$$ $$=$$ $$- 2^{9} \cdot 3^{3} \cdot 59$$ $$J_2$$ $$=$$ $$75$$ $$=$$ $$3 \cdot 5^{2}$$ $$J_4$$ $$=$$ $$126$$ $$=$$ $$2 \cdot 3^{2} \cdot 7$$ $$J_6$$ $$=$$ $$-1024$$ $$=$$ $$- 2^{10}$$ $$J_8$$ $$=$$ $$-23169$$ $$=$$ $$- 3 \cdot 7723$$ $$J_{10}$$ $$=$$ $$-6372$$ $$=$$ $$- 2^{2} \cdot 3^{3} \cdot 59$$ $$g_1$$ $$=$$ $$-87890625/236$$ $$g_2$$ $$=$$ $$-984375/118$$ $$g_3$$ $$=$$ $$160000/177$$

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
$$(1 : 0 : 0)$$ $$(1 : -1 : 0)$$ $$(0 : 0 : 1)$$ $$(-1 : 0 : 1)$$ $$(0 : -1 : 1)$$ $$(1 : 0 : 1)$$
$$(1 : -2 : 1)$$ $$(1 : -3 : 2)$$ $$(1 : -6 : 2)$$ $$(-3 : 70 : 5)$$ $$(-3 : -168 : 5)$$
All points
$$(1 : 0 : 0)$$ $$(1 : -1 : 0)$$ $$(0 : 0 : 1)$$ $$(-1 : 0 : 1)$$ $$(0 : -1 : 1)$$ $$(1 : 0 : 1)$$
$$(1 : -2 : 1)$$ $$(1 : -3 : 2)$$ $$(1 : -6 : 2)$$ $$(-3 : 70 : 5)$$ $$(-3 : -168 : 5)$$
All points
$$(1 : -1 : 0)$$ $$(1 : 1 : 0)$$ $$(-1 : 0 : 1)$$ $$(0 : -1 : 1)$$ $$(0 : 1 : 1)$$ $$(1 : -2 : 1)$$
$$(1 : 2 : 1)$$ $$(1 : -3 : 2)$$ $$(1 : 3 : 2)$$ $$(-3 : -238 : 5)$$ $$(-3 : 238 : 5)$$

magma: [C![-3,-168,5],C![-3,70,5],C![-1,0,1],C![0,-1,1],C![0,0,1],C![1,-6,2],C![1,-3,2],C![1,-2,1],C![1,-1,0],C![1,0,0],C![1,0,1]]; // minimal model

magma: [C![-3,-238,5],C![-3,238,5],C![-1,0,1],C![0,-1,1],C![0,1,1],C![1,-3,2],C![1,3,2],C![1,-2,1],C![1,-1,0],C![1,1,0],C![1,2,1]]; // 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: $$\Z \times \Z/{2}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(-1 : 0 : 1) + (0 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)$$ $$x (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-xz^2 - z^3$$ $$0.008698$$ $$\infty$$
$$D_0 - (1 : -1 : 0) - (1 : 0 : 0)$$ $$x^2 - xz + z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0$$ $$2$$
Generator $D_0$ Height Order
$$(-1 : 0 : 1) + (0 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)$$ $$x (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-xz^2 - z^3$$ $$0.008698$$ $$\infty$$
$$D_0 - (1 : -1 : 0) - (1 : 0 : 0)$$ $$x^2 - xz + z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0$$ $$2$$
Generator $D_0$ Height Order
$$(-1 : 0 : 1) + (0 : -1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)$$ $$x (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$x^3 - 2xz^2 - z^3$$ $$0.008698$$ $$\infty$$
$$D_0 - (1 : -1 : 0) - (1 : 1 : 0)$$ $$x^2 - xz + z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$x^3 + z^3$$ $$0$$ $$2$$

BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$1$$ Mordell-Weil rank: $$1$$ 2-Selmer rank: $$2$$ Regulator: $$0.008698$$ Real period: $$21.57586$$ Tamagawa product: $$4$$ Torsion order: $$2$$ Leading coefficient: $$0.187676$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$1$$ $$2$$ $$2$$ $$( 1 + T )( 1 + T + 2 T^{2} )$$
$$3$$ $$2$$ $$3$$ $$2$$ $$1 + 3 T + 3 T^{2}$$
$$59$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + 8 T + 59 T^{2} )$$

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 $$\Q$$.