Properties

 Label 610423.a.610423.1 Conductor 610423 Discriminant -610423 Mordell-Weil group $$\Z \times \Z$$ 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

 $y^2 + (x + 1)y = x^5 - 9x^3 + 7x^2 + 13x - 14$ (homogenize, simplify) $y^2 + (xz^2 + z^3)y = x^5z - 9x^3z^3 + 7x^2z^4 + 13xz^5 - 14z^6$ (dehomogenize, simplify) $y^2 = 4x^5 - 36x^3 + 29x^2 + 54x - 55$ (minimize, homogenize)

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-14, 13, 7, -9, 0, 1], R![1, 1]);

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-14, 13, 7, -9, 0, 1]), R([1, 1]));

magma: X,pi:= SimplifiedModel(C);

sage: X = HyperellipticCurve(R([-55, 54, 29, -36, 0, 4]))

Invariants

 Conductor: $$N$$ = $$610423$$ = $$11 \cdot 211 \cdot 263$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ = $$-610423$$ = $$- 11 \cdot 211 \cdot 263$$ magma: Discriminant(C); Factorization(Integers()!$1);

G2 invariants

 $$I_2$$ = $$16416$$ = $$2^{5} \cdot 3^{3} \cdot 19$$ $$I_4$$ = $$142464$$ = $$2^{7} \cdot 3 \cdot 7 \cdot 53$$ $$I_6$$ = $$802171584$$ = $$2^{6} \cdot 3^{2} \cdot 769 \cdot 1811$$ $$I_{10}$$ = $$-2500292608$$ = $$- 2^{12} \cdot 11 \cdot 211 \cdot 263$$ $$J_2$$ = $$2052$$ = $$2^{2} \cdot 3^{3} \cdot 19$$ $$J_4$$ = $$173962$$ = $$2 \cdot 86981$$ $$J_6$$ = $$19454065$$ = $$5 \cdot 89 \cdot 43717$$ $$J_8$$ = $$2414240984$$ = $$2^{3} \cdot 301780123$$ $$J_{10}$$ = $$-610423$$ = $$- 11 \cdot 211 \cdot 263$$ $$g_1$$ = $$-36382017816364032/610423$$ $$g_2$$ = $$-1503095107936896/610423$$ $$g_3$$ = $$-81915309311760/610423$$

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

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

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: $$\Z \times \Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$D_0 - 2 \cdot(1 : 0 : 0)$$ $$x^2 + xz - 4z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-2xz^2 + 2z^3$$ $$0.687204$$ $$\infty$$
$$D_0 - 2 \cdot(1 : 0 : 0)$$ $$x^2 - 2z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-xz^2$$ $$0.648883$$ $$\infty$$

BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$2$$ Mordell-Weil rank: $$2$$ 2-Selmer rank: $$2$$ Regulator: $$0.397075$$ Real period: $$11.34796$$ Tamagawa product: $$1$$ Torsion order: $$1$$ Leading coefficient: $$4.505994$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$11$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 - 2 T + 11 T^{2} )$$
$$211$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 - T + 211 T^{2} )$$
$$263$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + 18 T + 263 T^{2} )$$

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 $$\Q$$.