Properties

 Label 62411.b.62411.1 Conductor 62411 Discriminant -62411 Mordell-Weil group $$\Z \times \Z \times \Z$$ 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

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

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

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

magma: X,pi:= SimplifiedModel(C);

sage: X = HyperellipticCurve(R([1, -4, 0, 0, 0, 4]))

Invariants

 Conductor: $$N$$ = $$62411$$ = $$139 \cdot 449$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ = $$-62411$$ = $$- 139 \cdot 449$$ magma: Discriminant(C); Factorization(Integers()!$1);

G2 invariants

 $$I_2$$ = $$-640$$ = $$- 2^{7} \cdot 5$$ $$I_4$$ = $$-20480$$ = $$- 2^{12} \cdot 5$$ $$I_6$$ = $$1310720$$ = $$2^{18} \cdot 5$$ $$I_{10}$$ = $$-255635456$$ = $$- 2^{12} \cdot 139 \cdot 449$$ $$J_2$$ = $$-80$$ = $$- 2^{4} \cdot 5$$ $$J_4$$ = $$480$$ = $$2^{5} \cdot 3 \cdot 5$$ $$J_6$$ = $$1280$$ = $$2^{8} \cdot 5$$ $$J_8$$ = $$-83200$$ = $$- 2^{8} \cdot 5^{2} \cdot 13$$ $$J_{10}$$ = $$-62411$$ = $$- 139 \cdot 449$$ $$g_1$$ = $$3276800000/62411$$ $$g_2$$ = $$245760000/62411$$ $$g_3$$ = $$-8192000/62411$$

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

Known points
$$(1 : 0 : 0)$$ $$(0 : 0 : 1)$$ $$(-1 : 0 : 1)$$ $$(0 : -1 : 1)$$ $$(1 : 0 : 1)$$ $$(-1 : -1 : 1)$$
$$(1 : -1 : 1)$$ $$(2 : 5 : 1)$$ $$(2 : -6 : 1)$$ $$(3 : 15 : 1)$$ $$(3 : -16 : 1)$$ $$(1 : -30 : 4)$$
$$(1 : -34 : 4)$$ $$(-15 : 740 : 16)$$ $$(-15 : -4836 : 16)$$ $$(30 : 4929 : 1)$$ $$(30 : -4930 : 1)$$

magma: [C![-15,-4836,16],C![-15,740,16],C![-1,-1,1],C![-1,0,1],C![0,-1,1],C![0,0,1],C![1,-34,4],C![1,-30,4],C![1,-1,1],C![1,0,0],C![1,0,1],C![2,-6,1],C![2,5,1],C![3,-16,1],C![3,15,1],C![30,-4930,1],C![30,4929,1]];

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: $$\Z \times \Z \times \Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(1 : -1 : 1) - (1 : 0 : 0)$$ $$x - z$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-z^3$$ $$0.602986$$ $$\infty$$
$$(-1 : -1 : 1) + (0 : -1 : 1) - 2 \cdot(1 : 0 : 0)$$ $$x (x + z)$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-z^3$$ $$0.783665$$ $$\infty$$
$$(0 : -1 : 1) - (1 : 0 : 0)$$ $$x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-z^3$$ $$0.155861$$ $$\infty$$

BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$3$$   (upper bound) Mordell-Weil rank: $$3$$ 2-Selmer rank: $$3$$ Regulator: $$0.052702$$ Real period: $$14.17064$$ Tamagawa product: $$1$$ Torsion order: $$1$$ Leading coefficient: $$0.746826$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$139$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 + 19 T + 139 T^{2} )$$
$$449$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 - 5 T + 449 T^{2} )$$

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 $$\Q$$.