Properties

 Label 847.b.9317.1 Conductor 847 Discriminant 9317 Mordell-Weil group $$\Z/{10}\Z$$ Sato-Tate group $G_{3,3}$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R \times \R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\Q \times \Q$$ $$\overline{\Q}$$-simple no $$\mathrm{GL}_2$$-type yes

Related objects

Show commands for: SageMath / Magma

Simplified equation

 $y^2 + (x^2 + 1)y = x^5 + 2x^4 - 3x^3 + 2x^2 - x$ (homogenize, simplify) $y^2 + (x^2z + z^3)y = x^5z + 2x^4z^2 - 3x^3z^3 + 2x^2z^4 - xz^5$ (dehomogenize, simplify) $y^2 = 4x^5 + 9x^4 - 12x^3 + 10x^2 - 4x + 1$ (minimize, homogenize)

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

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

sage: X = HyperellipticCurve(R([1, -4, 10, -12, 9, 4]))

magma: X,pi:= SimplifiedModel(C);

Invariants

 Conductor: $$N$$ $$=$$ $$847$$ $$=$$ $$7 \cdot 11^{2}$$ magma: Conductor(LSeries(C)); Factorization($1); Discriminant: $$\Delta$$ $$=$$ $$9317$$ $$=$$ $$7 \cdot 11^{3}$$ magma: Discriminant(C); Factorization(Integers()!$1);

G2 invariants

 $$I_2$$ $$=$$ $$-1216$$ $$=$$ $$- 2^{6} \cdot 19$$ $$I_4$$ $$=$$ $$94912$$ $$=$$ $$2^{6} \cdot 1483$$ $$I_6$$ $$=$$ $$-28957760$$ $$=$$ $$- 2^{6} \cdot 5 \cdot 13 \cdot 6961$$ $$I_{10}$$ $$=$$ $$38162432$$ $$=$$ $$2^{12} \cdot 7 \cdot 11^{3}$$ $$J_2$$ $$=$$ $$-152$$ $$=$$ $$- 2^{3} \cdot 19$$ $$J_4$$ $$=$$ $$-26$$ $$=$$ $$- 2 \cdot 13$$ $$J_6$$ $$=$$ $$401$$ $$=$$ $$401$$ $$J_8$$ $$=$$ $$-15407$$ $$=$$ $$- 7 \cdot 31 \cdot 71$$ $$J_{10}$$ $$=$$ $$9317$$ $$=$$ $$7 \cdot 11^{3}$$ $$g_1$$ $$=$$ $$-81136812032/9317$$ $$g_2$$ $$=$$ $$91307008/9317$$ $$g_3$$ $$=$$ $$9264704/9317$$

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

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

Number of rational Weierstrass points: $$1$$

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: $$\Z/{10}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$D_0 - 2 \cdot(1 : 0 : 0)$$ $$x^2 + xz - z^2$$ $$=$$ $$0,$$ $$y$$ $$=$$ $$3xz^2 - 2z^3$$ $$0$$ $$10$$

BSD invariants

 Hasse-Weil conjecture: verified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$1$$ Regulator: $$1$$ Real period: $$16.82727$$ Tamagawa product: $$2$$ Torsion order: $$10$$ Leading coefficient: $$0.336545$$ Analytic order of Ш: $$1$$   (rounded) Order of Ш: square

Local invariants

Prime ord($$N$$) ord($$\Delta$$) Tamagawa L-factor Cluster picture
$$7$$ $$1$$ $$1$$ $$1$$ $$( 1 + T )( 1 + 2 T + 7 T^{2} )$$
$$11$$ $$2$$ $$3$$ $$2$$ $$( 1 - T )^{2}$$

Sato-Tate group

 $$\mathrm{ST}$$ $$\simeq$$ $G_{3,3}$ $$\mathrm{ST}^0$$ $$\simeq$$ $$\mathrm{SU}(2)\times\mathrm{SU}(2)$$

Decomposition of the Jacobian

Splits over $$\Q$$

Decomposes up to isogeny as the product of the non-isogenous elliptic curves:
Elliptic curve 77.c2
Elliptic curve 11.a3

Endomorphisms of the Jacobian

Of $$\GL_2$$-type over $$\Q$$

Endomorphism ring over $$\Q$$:

 $$\End (J_{})$$ $$\simeq$$ an order of index $$3$$ in $$\Z \times \Z$$ $$\End (J_{}) \otimes \Q$$ $$\simeq$$ $$\Q$$ $$\times$$ $$\Q$$ $$\End (J_{}) \otimes \R$$ $$\simeq$$ $$\R \times \R$$

All $$\overline{\Q}$$-endomorphisms of the Jacobian are defined over $$\Q$$.