Properties

 Label 100240.a.400960.1 Conductor 100240 Discriminant 400960 Mordell-Weil group $$\Z$$ 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

 $y^2 + x^3y = x^5 - x^4 - 5x^3 + 6x^2 + 10x - 15$ (homogenize, simplify) $y^2 + x^3y = x^5z - x^4z^2 - 5x^3z^3 + 6x^2z^4 + 10xz^5 - 15z^6$ (dehomogenize, simplify) $y^2 = x^6 + 4x^5 - 4x^4 - 20x^3 + 24x^2 + 40x - 60$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-15, 10, 6, -5, -1, 1]), R([0, 0, 0, 1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-15, 10, 6, -5, -1, 1], R![0, 0, 0, 1]);

sage: X = HyperellipticCurve(R([-60, 40, 24, -20, -4, 4, 1]))

magma: X,pi:= SimplifiedModel(C);

Invariants

 Conductor: $$N$$ $$=$$ $$100240$$ $$=$$ $$2^{4} \cdot 5 \cdot 7 \cdot 179$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$400960$$ $$=$$ $$2^{6} \cdot 5 \cdot 7 \cdot 179$$ magma: Discriminant(C); Factorization(Integers()!$1);

G2 invariants

 $$I_2$$ $$=$$ $$24736$$ $$=$$ $$2^{5} \cdot 773$$ $$I_4$$ $$=$$ $$202816$$ $$=$$ $$2^{6} \cdot 3169$$ $$I_6$$ $$=$$ $$1664700928$$ $$=$$ $$2^{9} \cdot 11 \cdot 17 \cdot 17387$$ $$I_{10}$$ $$=$$ $$1642332160$$ $$=$$ $$2^{18} \cdot 5 \cdot 7 \cdot 179$$ $$J_2$$ $$=$$ $$3092$$ $$=$$ $$2^{2} \cdot 773$$ $$J_4$$ $$=$$ $$396240$$ $$=$$ $$2^{4} \cdot 3 \cdot 5 \cdot 13 \cdot 127$$ $$J_6$$ $$=$$ $$67352576$$ $$=$$ $$2^{11} \cdot 32887$$ $$J_8$$ $$=$$ $$12812006848$$ $$=$$ $$2^{6} \cdot 23 \cdot 103 \cdot 84503$$ $$J_{10}$$ $$=$$ $$400960$$ $$=$$ $$2^{6} \cdot 5 \cdot 7 \cdot 179$$ $$g_1$$ $$=$$ $$4415881923441488/6265$$ $$g_2$$ $$=$$ $$36603852142416/1253$$ $$g_3$$ $$=$$ $$10061279346176/6265$$

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: $$(1 : 0 : 0),\, (1 : -1 : 0),\, (-3 : 12 : 1),\, (-3 : 15 : 1)$$

magma: [C![-3,12,1],C![-3,15,1],C![1,-1,0],C![1,0,0]];

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: $$\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$D_0 - (1 : -1 : 0) - (1 : 0 : 0)$$ $$x^2 + xz - 4z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-3xz^2 + z^3$$ $$0.061812$$ $$\infty$$

BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$1$$ Mordell-Weil rank: $$1$$ 2-Selmer rank: $$1$$ Regulator: $$0.061812$$ Real period: $$6.806168$$ Tamagawa product: $$5$$ Torsion order: $$1$$ Leading coefficient: $$2.103531$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor
$$2$$ $$4$$ $$6$$ $$5$$ $$1$$
$$5$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + T + 5 T^{2} )$$
$$7$$ $$1$$ $$1$$ $$1$$ $$( 1 - T )( 1 + 2 T + 7 T^{2} )$$
$$179$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 + 24 T + 179 T^{2} )$$

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 $$\Q$$.