Properties

 Label 797.a.797.1 Conductor 797 Discriminant 797 Mordell-Weil group $$\Z/{7}\Z$$ 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

Learn more about

Show commands for: SageMath / Magma

Simplified equation

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

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

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

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

magma: X,pi:= SimplifiedModel(C);

Invariants

 Conductor: $$N$$ $$=$$ $$797$$ $$=$$ $$797$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$797$$ $$=$$ $$797$$ magma: Discriminant(C); Factorization(Integers()!$1);

G2 invariants

 $$I_2$$ $$=$$ $$96$$ $$=$$ $$2^{5} \cdot 3$$ $$I_4$$ $$=$$ $$8448$$ $$=$$ $$2^{8} \cdot 3 \cdot 11$$ $$I_6$$ $$=$$ $$486912$$ $$=$$ $$2^{9} \cdot 3 \cdot 317$$ $$I_{10}$$ $$=$$ $$3264512$$ $$=$$ $$2^{12} \cdot 797$$ $$J_2$$ $$=$$ $$12$$ $$=$$ $$2^{2} \cdot 3$$ $$J_4$$ $$=$$ $$-82$$ $$=$$ $$- 2 \cdot 41$$ $$J_6$$ $$=$$ $$-548$$ $$=$$ $$- 2^{2} \cdot 137$$ $$J_8$$ $$=$$ $$-3325$$ $$=$$ $$- 5^{2} \cdot 7 \cdot 19$$ $$J_{10}$$ $$=$$ $$797$$ $$=$$ $$797$$ $$g_1$$ $$=$$ $$248832/797$$ $$g_2$$ $$=$$ $$-141696/797$$ $$g_3$$ $$=$$ $$-78912/797$$

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

magma: [C![0,-1,1],C![0,0,1],C![1,0,0]];

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: $$\Z/{7}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$D_0 - 2 \cdot(1 : 0 : 0)$$ $$x^2 - xz + z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0$$ $$7$$

BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$0$$ Regulator: $$1$$ Real period: $$17.44098$$ Tamagawa product: $$1$$ Torsion order: $$7$$ Leading coefficient: $$0.355938$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$797$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 + 14 T + 797 T^{2} )$$

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 $$\Q$$.