Properties

 Label 1472.a.5888.1 Conductor 1472 Discriminant -5888 Mordell-Weil group $$\Z/{2}\Z \times \Z/{4}\Z$$ 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

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Show commands for: SageMath / Magma

Simplified equation

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

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

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

sage: X = HyperellipticCurve(R([0, -1, -2, -1, 1, 1]))

magma: X,pi:= SimplifiedModel(C);

Invariants

 Conductor: $$N$$ $$=$$ $$1472$$ $$=$$ $$2^{6} \cdot 23$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$-5888$$ $$=$$ $$- 2^{8} \cdot 23$$ magma: Discriminant(C); Factorization(Integers()!$1);

G2 invariants

 $$I_2$$ $$=$$ $$-32$$ $$=$$ $$- 2^{5}$$ $$I_4$$ $$=$$ $$-14336$$ $$=$$ $$- 2^{11} \cdot 7$$ $$I_6$$ $$=$$ $$-303104$$ $$=$$ $$- 2^{13} \cdot 37$$ $$I_{10}$$ $$=$$ $$-24117248$$ $$=$$ $$- 2^{20} \cdot 23$$ $$J_2$$ $$=$$ $$-4$$ $$=$$ $$- 2^{2}$$ $$J_4$$ $$=$$ $$150$$ $$=$$ $$2 \cdot 3 \cdot 5^{2}$$ $$J_6$$ $$=$$ $$692$$ $$=$$ $$2^{2} \cdot 173$$ $$J_8$$ $$=$$ $$-6317$$ $$=$$ $$-6317$$ $$J_{10}$$ $$=$$ $$-5888$$ $$=$$ $$- 2^{8} \cdot 23$$ $$g_1$$ $$=$$ $$4/23$$ $$g_2$$ $$=$$ $$75/46$$ $$g_3$$ $$=$$ $$-173/92$$

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

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

Number of rational Weierstrass points: $$3$$

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: $$\Z/{2}\Z \times \Z/{4}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

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

BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$2$$ Regulator: $$1$$ Real period: $$15.71763$$ Tamagawa product: $$2$$ Torsion order: $$8$$ Leading coefficient: $$0.491176$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$6$$ $$8$$ $$2$$ $$1$$
$$23$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 - 8 T + 23 T^{2} )$$

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 $$\Q$$.