Properties

 Label 997.b.997.1 Conductor 997 Discriminant 997 Mordell-Weil group $$\Z \times \Z/{3}\Z$$ 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

The Jacobian of this curve is an apparently paramodular abelian surface appearing as entry "997b" in the table of Brumer and Kramer [MR:3165645].

Simplified equation

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

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

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

sage: X = HyperellipticCurve(R([1, 0, -4, 8, -8, 4]))

magma: X,pi:= SimplifiedModel(C);

Invariants

 Conductor: $$N$$ $$=$$ $$997$$ $$=$$ $$997$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$997$$ $$=$$ $$997$$ magma: Discriminant(C); Factorization(Integers()!$1);

G2 invariants

 $$I_2$$ $$=$$ $$-128$$ $$=$$ $$- 2^{7}$$ $$I_4$$ $$=$$ $$256$$ $$=$$ $$2^{8}$$ $$I_6$$ $$=$$ $$107520$$ $$=$$ $$2^{10} \cdot 3 \cdot 5 \cdot 7$$ $$I_{10}$$ $$=$$ $$4083712$$ $$=$$ $$2^{12} \cdot 997$$ $$J_2$$ $$=$$ $$-16$$ $$=$$ $$- 2^{4}$$ $$J_4$$ $$=$$ $$8$$ $$=$$ $$2^{3}$$ $$J_6$$ $$=$$ $$-208$$ $$=$$ $$- 2^{4} \cdot 13$$ $$J_8$$ $$=$$ $$816$$ $$=$$ $$2^{4} \cdot 3 \cdot 17$$ $$J_{10}$$ $$=$$ $$997$$ $$=$$ $$997$$ $$g_1$$ $$=$$ $$-1048576/997$$ $$g_2$$ $$=$$ $$-32768/997$$ $$g_3$$ $$=$$ $$-53248/997$$

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: $$(1 : 0 : 0),\, (0 : 0 : 1),\, (0 : -1 : 1),\, (1 : 0 : 1),\, (1 : -1 : 1),\, (2 : 3 : 1),\, (2 : -4 : 1)$$

magma: [C![0,-1,1],C![0,0,1],C![1,-1,1],C![1,0,0],C![1,0,1],C![2,-4,1],C![2,3,1]];

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: $$\Z \times \Z/{3}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(0 : -1 : 1) - (1 : 0 : 0)$$ $$x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-z^3$$ $$0.081269$$ $$\infty$$
$$(0 : 0 : 1) + (1 : 0 : 1) - 2 \cdot(1 : 0 : 0)$$ $$(-x + z) x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0$$ $$3$$

BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$1$$ Mordell-Weil rank: $$1$$ 2-Selmer rank: $$1$$ Regulator: $$0.081269$$ Real period: $$19.93284$$ Tamagawa product: $$1$$ Torsion order: $$3$$ Leading coefficient: $$0.179992$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$997$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + 28 T + 997 T^{2} )$$

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 $$\Q$$.