Properties

 Label 2512.a.160768.1 Conductor 2512 Discriminant -160768 Mordell-Weil group $$\Z/{13}\Z$$ 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

 $y^2 + (x + 1)y = x^6 + 3x^4 + x^3 + 2x^2$ (homogenize, simplify) $y^2 + (xz^2 + z^3)y = x^6 + 3x^4z^2 + x^3z^3 + 2x^2z^4$ (dehomogenize, simplify) $y^2 = 4x^6 + 12x^4 + 4x^3 + 9x^2 + 2x + 1$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 0, 2, 1, 3, 0, 1]), R([1, 1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 0, 2, 1, 3, 0, 1], R![1, 1]);

sage: X = HyperellipticCurve(R([1, 2, 9, 4, 12, 0, 4]))

magma: X,pi:= SimplifiedModel(C);

Invariants

 Conductor: $$N$$ $$=$$ $$2512$$ $$=$$ $$2^{4} \cdot 157$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$-160768$$ $$=$$ $$- 2^{10} \cdot 157$$ magma: Discriminant(C); Factorization(Integers()!$1);

G2 invariants

 $$I_2$$ $$=$$ $$-2592$$ $$=$$ $$- 2^{5} \cdot 3^{4}$$ $$I_4$$ $$=$$ $$90432$$ $$=$$ $$2^{6} \cdot 3^{2} \cdot 157$$ $$I_6$$ $$=$$ $$-74285568$$ $$=$$ $$- 2^{9} \cdot 3^{2} \cdot 7^{3} \cdot 47$$ $$I_{10}$$ $$=$$ $$-658505728$$ $$=$$ $$- 2^{22} \cdot 157$$ $$J_2$$ $$=$$ $$-324$$ $$=$$ $$- 2^{2} \cdot 3^{4}$$ $$J_4$$ $$=$$ $$3432$$ $$=$$ $$2^{3} \cdot 3 \cdot 11 \cdot 13$$ $$J_6$$ $$=$$ $$-34544$$ $$=$$ $$- 2^{4} \cdot 17 \cdot 127$$ $$J_8$$ $$=$$ $$-146592$$ $$=$$ $$- 2^{5} \cdot 3^{2} \cdot 509$$ $$J_{10}$$ $$=$$ $$-160768$$ $$=$$ $$- 2^{10} \cdot 157$$ $$g_1$$ $$=$$ $$3486784401/157$$ $$g_2$$ $$=$$ $$227988189/314$$ $$g_3$$ $$=$$ $$14165199/628$$

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

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

Number of rational Weierstrass points: $$0$$

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: $$\Z/{13}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(0 : -1 : 1) - (1 : -1 : 0)$$ $$z x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$x^3 - z^3$$ $$0$$ $$13$$

BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$0$$ Regulator: $$1$$ Real period: $$10.60147$$ Tamagawa product: $$13$$ Torsion order: $$13$$ Leading coefficient: $$0.815498$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$4$$ $$10$$ $$13$$ $$1 - T$$
$$157$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 + 11 T + 157 T^{2} )$$

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 $$\Q$$.