Properties

 Label 100312.a.200624.1 Conductor 100312 Discriminant -200624 Mordell-Weil group $$\Z/{2}\Z$$ 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

 $y^2 + (x^2 + 1)y = -x^6 - 2x^5 + x^4 + 3x^3 - x - 1$ (homogenize, simplify) $y^2 + (x^2z + z^3)y = -x^6 - 2x^5z + x^4z^2 + 3x^3z^3 - xz^5 - z^6$ (dehomogenize, simplify) $y^2 = -4x^6 - 8x^5 + 5x^4 + 12x^3 + 2x^2 - 4x - 3$ (minimize, homogenize)

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

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

magma: X,pi:= SimplifiedModel(C);

sage: X = HyperellipticCurve(R([-3, -4, 2, 12, 5, -8, -4]))

Invariants

 Conductor: $$N$$ = $$100312$$ = $$2^{3} \cdot 12539$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ = $$-200624$$ = $$- 2^{4} \cdot 12539$$ magma: Discriminant(C); Factorization(Integers()!$1);

G2 invariants

 $$I_2$$ = $$-896$$ = $$- 2^{7} \cdot 7$$ $$I_4$$ = $$70912$$ = $$2^{8} \cdot 277$$ $$I_6$$ = $$-86923520$$ = $$- 2^{8} \cdot 5 \cdot 59 \cdot 1151$$ $$I_{10}$$ = $$-821755904$$ = $$- 2^{16} \cdot 12539$$ $$J_2$$ = $$-112$$ = $$- 2^{4} \cdot 7$$ $$J_4$$ = $$-216$$ = $$- 2^{3} \cdot 3^{3}$$ $$J_6$$ = $$124676$$ = $$2^{2} \cdot 71 \cdot 439$$ $$J_8$$ = $$-3502592$$ = $$- 2^{9} \cdot 6841$$ $$J_{10}$$ = $$-200624$$ = $$- 2^{4} \cdot 12539$$ $$g_1$$ = $$1101463552/12539$$ $$g_2$$ = $$-18966528/12539$$ $$g_3$$ = $$-97745984/12539$$

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

All points: $$(-1 : -1 : 1),\, (1 : -1 : 1)$$

magma: [C![-1,-1,1],C![1,-1,1]];

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: $$\Z/{2}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

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

BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$1$$ Regulator: $$1$$ Real period: $$5.298723$$ Tamagawa product: $$2$$ Torsion order: $$2$$ Leading coefficient: $$2.649361$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$2$$ $$4$$ $$3$$ $$2$$ $$1 - T$$
$$12539$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 - 40 T + 12539 T^{2} )$$

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 $$\Q$$.