Properties

 Label 1051.a.1051.1 Conductor $1051$ Discriminant $-1051$ Mordell-Weil group $$\Z$$ 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 for: SageMath / Magma

Simplified equation

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

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

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

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

magma: X,pi:= SimplifiedModel(C);

Invariants

 Conductor: $$N$$ $$=$$ $$1051$$ $$=$$ $$1051$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$-1051$$ $$=$$ $$-1051$$ magma: Discriminant(C); Factorization(Integers()!$1);

G2 invariants

 $$I_2$$ $$=$$ $$96$$ $$=$$ $$2^{5} \cdot 3$$ $$I_4$$ $$=$$ $$-144$$ $$=$$ $$- 2^{4} \cdot 3^{2}$$ $$I_6$$ $$=$$ $$144$$ $$=$$ $$2^{4} \cdot 3^{2}$$ $$I_{10}$$ $$=$$ $$4204$$ $$=$$ $$2^{2} \cdot 1051$$ $$J_2$$ $$=$$ $$48$$ $$=$$ $$2^{4} \cdot 3$$ $$J_4$$ $$=$$ $$120$$ $$=$$ $$2^{3} \cdot 3 \cdot 5$$ $$J_6$$ $$=$$ $$-80$$ $$=$$ $$- 2^{4} \cdot 5$$ $$J_8$$ $$=$$ $$-4560$$ $$=$$ $$- 2^{4} \cdot 3 \cdot 5 \cdot 19$$ $$J_{10}$$ $$=$$ $$1051$$ $$=$$ $$1051$$ $$g_1$$ $$=$$ $$254803968/1051$$ $$g_2$$ $$=$$ $$13271040/1051$$ $$g_3$$ $$=$$ $$-184320/1051$$

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

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

magma: [C![-1,-1,1],C![-1,1,1],C![0,-1,1],C![0,1,1],C![1,-1,1],C![1,0,0],C![1,1,1]]; // 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: $$\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(0 : -1 : 1) - (1 : 0 : 0)$$ $$x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-z^3$$ $$0.007924$$ $$\infty$$
Generator $D_0$ Height Order
$$(0 : -1 : 1) - (1 : 0 : 0)$$ $$x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-z^3$$ $$0.007924$$ $$\infty$$
Generator $D_0$ Height Order
$$(0 : -1 : 1) - (1 : 0 : 0)$$ $$x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-z^3$$ $$0.007924$$ $$\infty$$

BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$1$$ Mordell-Weil rank: $$1$$ 2-Selmer rank: $$1$$ Regulator: $$0.007924$$ Real period: $$23.43782$$ Tamagawa product: $$1$$ Torsion order: $$1$$ Leading coefficient: $$0.185743$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$1051$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + 36 T + 1051 T^{2} )$$

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 $$\Q$$.