Properties

 Label 7200.d.345600.1 Conductor 7200 Discriminant 345600 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

 $y^2 + y = x^6 - 2x^4 - x^3 + 4x^2 - 2x$ (homogenize, simplify) $y^2 + z^3y = x^6 - 2x^4z^2 - x^3z^3 + 4x^2z^4 - 2xz^5$ (dehomogenize, simplify) $y^2 = 4x^6 - 8x^4 - 4x^3 + 16x^2 - 8x + 1$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -2, 4, -1, -2, 0, 1]), R([1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -2, 4, -1, -2, 0, 1], R![1]);

sage: X = HyperellipticCurve(R([1, -8, 16, -4, -8, 0, 4]))

magma: X,pi:= SimplifiedModel(C);

Invariants

 Conductor: $$N$$ $$=$$ $$7200$$ $$=$$ $$2^{5} \cdot 3^{2} \cdot 5^{2}$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$345600$$ $$=$$ $$2^{9} \cdot 3^{3} \cdot 5^{2}$$ magma: Discriminant(C); Factorization(Integers()!$1);

G2 invariants

 $$I_2$$ $$=$$ $$1184$$ $$=$$ $$2^{5} \cdot 37$$ $$I_4$$ $$=$$ $$124480$$ $$=$$ $$2^{6} \cdot 5 \cdot 389$$ $$I_6$$ $$=$$ $$29719040$$ $$=$$ $$2^{9} \cdot 5 \cdot 13 \cdot 19 \cdot 47$$ $$I_{10}$$ $$=$$ $$1415577600$$ $$=$$ $$2^{21} \cdot 3^{3} \cdot 5^{2}$$ $$J_2$$ $$=$$ $$148$$ $$=$$ $$2^{2} \cdot 37$$ $$J_4$$ $$=$$ $$-384$$ $$=$$ $$- 2^{7} \cdot 3$$ $$J_6$$ $$=$$ $$9216$$ $$=$$ $$2^{10} \cdot 3^{2}$$ $$J_8$$ $$=$$ $$304128$$ $$=$$ $$2^{10} \cdot 3^{3} \cdot 11$$ $$J_{10}$$ $$=$$ $$345600$$ $$=$$ $$2^{9} \cdot 3^{3} \cdot 5^{2}$$ $$g_1$$ $$=$$ $$138687914/675$$ $$g_2$$ $$=$$ $$-810448/225$$ $$g_3$$ $$=$$ $$43808/75$$

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
$$(1 : -1 : 0)$$ $$(1 : 1 : 0)$$ $$(0 : 0 : 1)$$ $$(0 : -1 : 1)$$ $$(1 : 0 : 1)$$ $$(1 : -1 : 1)$$
$$(-1 : 2 : 1)$$ $$(-1 : -3 : 1)$$ $$(1 : -3 : 2)$$ $$(1 : -5 : 2)$$

magma: [C![-1,-3,1],C![-1,2,1],C![0,-1,1],C![0,0,1],C![1,-5,2],C![1,-3,2],C![1,-1,0],C![1,-1,1],C![1,0,1],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: $$\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$(0 : 0 : 1) - (1 : 1 : 0)$$ $$z x$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$-x^3$$ $$0.001909$$ $$\infty$$

BSD invariants

 Hasse-Weil conjecture: unverified Analytic rank: $$1$$ Mordell-Weil rank: $$1$$ 2-Selmer rank: $$1$$ Regulator: $$0.001909$$ Real period: $$17.24456$$ Tamagawa product: $$24$$ Torsion order: $$1$$ Leading coefficient: $$0.790323$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$2$$ $$5$$ $$9$$ $$8$$ $$1$$
$$3$$ $$2$$ $$3$$ $$3$$ $$1 + T + T^{2}$$
$$5$$ $$2$$ $$2$$ $$1$$ $$1 + 4 T + 5 T^{2}$$

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 $$\Q$$.