Properties

 Label 709.a.709.1 Conductor 709 Discriminant 709 Mordell-Weil group $$\Z/{8}\Z$$ 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

Simplified equation

 $y^2 + xy = x^5 - 2x^2 + x$ (homogenize, simplify) $y^2 + xz^2y = x^5z - 2x^2z^4 + xz^5$ (dehomogenize, simplify) $y^2 = 4x^5 - 7x^2 + 4x$ (minimize, homogenize)

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

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

sage: X = HyperellipticCurve(R([0, 4, -7, 0, 0, 4]))

magma: X,pi:= SimplifiedModel(C);

Invariants

 Conductor: $$N$$ $$=$$ $$709$$ $$=$$ $$709$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$709$$ $$=$$ $$709$$ magma: Discriminant(C); Factorization(Integers()!$1);

G2 invariants

 $$I_2$$ $$=$$ $$640$$ $$=$$ $$2^{7} \cdot 5$$ $$I_4$$ $$=$$ $$-20480$$ $$=$$ $$- 2^{12} \cdot 5$$ $$I_6$$ $$=$$ $$-2693696$$ $$=$$ $$- 2^{6} \cdot 42089$$ $$I_{10}$$ $$=$$ $$2904064$$ $$=$$ $$2^{12} \cdot 709$$ $$J_2$$ $$=$$ $$80$$ $$=$$ $$2^{4} \cdot 5$$ $$J_4$$ $$=$$ $$480$$ $$=$$ $$2^{5} \cdot 3 \cdot 5$$ $$J_6$$ $$=$$ $$1121$$ $$=$$ $$19 \cdot 59$$ $$J_8$$ $$=$$ $$-35180$$ $$=$$ $$- 2^{2} \cdot 5 \cdot 1759$$ $$J_{10}$$ $$=$$ $$709$$ $$=$$ $$709$$ $$g_1$$ $$=$$ $$3276800000/709$$ $$g_2$$ $$=$$ $$245760000/709$$ $$g_3$$ $$=$$ $$7174400/709$$

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

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

Number of rational Weierstrass points: $$2$$

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: $$\Z/{8}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(1 : 0 : 1) - (1 : 0 : 0)$$ $$x - z$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$0$$ $$0$$ $$8$$

BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$1$$ Regulator: $$1$$ Real period: $$18.36116$$ Tamagawa product: $$1$$ Torsion order: $$8$$ Leading coefficient: $$0.286893$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$709$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 - 6 T + 709 T^{2} )$$

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 $$\Q$$.