Properties

 Label 847.d.847.1 Conductor $847$ Discriminant $-847$ Mordell-Weil group $$\Z/{3}\Z$$ Sato-Tate group $\mathrm{SU}(2)\times\mathrm{SU}(2)$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\R \times \R$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\Q \times \Q$$ $$\End(J) \otimes \Q$$ $$\Q \times \Q$$ $$\overline{\Q}$$-simple no $$\mathrm{GL}_2$$-type yes

Related objects

Show commands for: SageMath / Magma

Simplified equation

 $y^2 + (x^3 + x^2 + x + 1)y = -12x^6 - 15x^5 + 9x^4 + 31x^3 + 9x^2 - 15x - 12$ (homogenize, simplify) $y^2 + (x^3 + x^2z + xz^2 + z^3)y = -12x^6 - 15x^5z + 9x^4z^2 + 31x^3z^3 + 9x^2z^4 - 15xz^5 - 12z^6$ (dehomogenize, simplify) $y^2 = -47x^6 - 58x^5 + 39x^4 + 128x^3 + 39x^2 - 58x - 47$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-12, -15, 9, 31, 9, -15, -12]), R([1, 1, 1, 1]));

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-12, -15, 9, 31, 9, -15, -12], R![1, 1, 1, 1]);

sage: X = HyperellipticCurve(R([-47, -58, 39, 128, 39, -58, -47]))

magma: X,pi:= SimplifiedModel(C);

Invariants

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

G2 invariants

 $$I_2$$ $$=$$ $$80408$$ $$=$$ $$2^{3} \cdot 19 \cdot 23^{2}$$ $$I_4$$ $$=$$ $$402403732$$ $$=$$ $$2^{2} \cdot 2441 \cdot 41213$$ $$I_6$$ $$=$$ $$8094753026048$$ $$=$$ $$2^{12} \cdot 13 \cdot 67 \cdot 499 \cdot 4547$$ $$I_{10}$$ $$=$$ $$3388$$ $$=$$ $$2^{2} \cdot 7 \cdot 11^{2}$$ $$J_2$$ $$=$$ $$40204$$ $$=$$ $$2^{2} \cdot 19 \cdot 23^{2}$$ $$J_4$$ $$=$$ $$281112$$ $$=$$ $$2^{3} \cdot 3 \cdot 13 \cdot 17 \cdot 53$$ $$J_6$$ $$=$$ $$1967560$$ $$=$$ $$2^{3} \cdot 5 \cdot 7 \cdot 7027$$ $$J_8$$ $$=$$ $$19956424$$ $$=$$ $$2^{3} \cdot 2494553$$ $$J_{10}$$ $$=$$ $$847$$ $$=$$ $$7 \cdot 11^{2}$$ $$g_1$$ $$=$$ $$105037970421355597057024/847$$ $$g_2$$ $$=$$ $$18267839107785466368/847$$ $$g_3$$ $$=$$ $$454326923025280/121$$

sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]

magma: IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);

Automorphism group

 $$\mathrm{Aut}(X)$$ $$\simeq$$ $C_2^2$ magma: AutomorphismGroup(C); IdentifyGroup($1); $$\mathrm{Aut}(X_{\overline{\Q}})$$ $$\simeq$$$C_2^2$magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);

Rational points

 This curve has no rational points. This curve has no rational points. This curve has no rational points.

magma: []; // minimal model

magma: []; // simplified model

Number of rational Weierstrass points: $$0$$

magma: #Roots(HyperellipticPolynomials(SimplifiedModel(C)));

This curve is locally solvable except over $\R$.

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: $$\Z/{3}\Z$$

magma: MordellWeilGroupGenus2(Jacobian(C));

Generator $D_0$ Height Order
$$D_0 - D_\infty$$ $$18x^2 + 29xz + 18z^2$$ $$=$$ $$0,$$ $$162y$$ $$=$$ $$-77xz^2 - 126z^3$$ $$0$$ $$3$$
Generator $D_0$ Height Order
$$D_0 - D_\infty$$ $$18x^2 + 29xz + 18z^2$$ $$=$$ $$0,$$ $$162y$$ $$=$$ $$-77xz^2 - 126z^3$$ $$0$$ $$3$$
Generator $D_0$ Height Order
$$D_0 - D_\infty$$ $$18x^2 + 29xz + 18z^2$$ $$=$$ $$0,$$ $$162y$$ $$=$$ $$x^3 + x^2z - 153xz^2 - 251z^3$$ $$0$$ $$3$$

BSD invariants

 Hasse-Weil conjecture: verified Analytic rank: $$0$$ Mordell-Weil rank: $$0$$ 2-Selmer rank: $$1$$ Regulator: $$1$$ Real period: $$1.179534$$ Tamagawa product: $$1$$ Torsion order: $$3$$ Leading coefficient: $$0.262118$$ Analytic order of Ш: $$2$$   (rounded) Order of Ш: twice a square

Local invariants

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

Sato-Tate group

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

Decomposition of the Jacobian

Splits over $$\Q$$

Decomposes up to isogeny as the product of the non-isogenous elliptic curves:
Elliptic curve 77.b1
Elliptic curve 11.a1

Endomorphisms of the Jacobian

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

Endomorphism ring over $$\Q$$:

 $$\End (J_{})$$ $$\simeq$$ an order of index $$2$$ 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 $$\Q$$.