Properties

 Label 3969.b.35721.1 Conductor 3969 Discriminant 35721 Sato-Tate group $E_6$ $$\End(J_{\overline{\Q}}) \otimes \R$$ $$\mathrm{M}_2(\R)$$ $$\End(J_{\overline{\Q}}) \otimes \Q$$ $$\mathrm{M}_2(\Q)$$ $$\overline{\Q}$$-simple no $$\mathrm{GL}_2$$-type yes

Related objects

Show commands for: Magma / SageMath

Minimal equation

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

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

$y^2 + (x^3 + x + 1)y = -2x^5 + 3x^4 - 3x^2$

Invariants

 magma: Conductor(LSeries(C)); Factorization($1); $$N$$ = $$3969$$ = $$3^{4} \cdot 7^{2}$$ magma: Discriminant(C); Factorization(Integers()!$1); $$\Delta$$ = $$35721$$ = $$3^{6} \cdot 7^{2}$$

G2 invariants

magma: G2Invariants(C);

 $$I_2$$ = $$1608$$ = $$2^{3} \cdot 3 \cdot 67$$ $$I_4$$ = $$106596$$ = $$2^{2} \cdot 3^{4} \cdot 7 \cdot 47$$ $$I_6$$ = $$46861416$$ = $$2^{3} \cdot 3^{4} \cdot 7 \cdot 10331$$ $$I_{10}$$ = $$146313216$$ = $$2^{12} \cdot 3^{6} \cdot 7^{2}$$ $$J_2$$ = $$201$$ = $$3 \cdot 67$$ $$J_4$$ = $$573$$ = $$3 \cdot 191$$ $$J_6$$ = $$-563$$ = $$-1 \cdot 563$$ $$J_8$$ = $$-110373$$ = $$-1 \cdot 3 \cdot 36791$$ $$J_{10}$$ = $$35721$$ = $$3^{6} \cdot 7^{2}$$ $$g_1$$ = $$1350125107/147$$ $$g_2$$ = $$57445733/441$$ $$g_3$$ = $$-2527307/3969$$
Alternative geometric invariants: G2

Automorphism group

 magma: AutomorphismGroup(C); IdentifyGroup($1); $$\mathrm{Aut}(X)$$ $$\simeq$$ $$C_6$$ (GAP id : [6,2]) magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1); $$\mathrm{Aut}(X_{\overline{\Q}})$$ $$\simeq$$ $$D_6$$ (GAP id : [12,4])

Rational points

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);

This curve is locally solvable everywhere.

magma: [C![-5,-46,1],C![-5,175,1],C![-1,-1,1],C![-1,2,1],C![0,-1,1],C![0,0,1],C![1,-237,6],C![1,-16,6],C![1,-8,2],C![1,-5,2],C![1,-2,1],C![1,-1,0],C![1,-1,1],C![1,0,0],C![2,-7,1],C![2,-4,1],C![6,-356,5],C![6,-135,5]];

Known rational points: (-5 : -46 : 1), (-5 : 175 : 1), (-1 : -1 : 1), (-1 : 2 : 1), (0 : -1 : 1), (0 : 0 : 1), (1 : -237 : 6), (1 : -16 : 6), (1 : -8 : 2), (1 : -5 : 2), (1 : -2 : 1), (1 : -1 : 0), (1 : -1 : 1), (1 : 0 : 0), (2 : -7 : 1), (2 : -4 : 1), (6 : -356 : 5), (6 : -135 : 5)

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

Number of rational Weierstrass points: $$0$$

Invariants of the Jacobian:

Analytic rank*: $$2$$

magma: TwoSelmerGroup(Jacobian(C)); NumberOfGenerators($1); 2-Selmer rank: $$2$$ magma: HasSquareSha(Jacobian(C)); Order of Ш*: square Regulator: 0.00315548079438 Real period: 23.234166964744789092793173025 Tamagawa numbers: 3 (p = 3), 1 (p = 7) magma: TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);

Torsion: $$\mathrm{trivial}$$

Sato-Tate group

 $$\mathrm{ST}$$ $$\simeq$$ $E_6$ $$\mathrm{ST}^0$$ $$\simeq$$ $$\mathrm{SU}(2)$$

Decomposition

Splits over the number field $$\Q (b) \simeq$$ 6.6.330812181.1 with defining polynomial:
$$x^{6} - 21 x^{4} - 14 x^{3} + 63 x^{2} + 21 x - 35$$

Decomposes up to isogeny as the square of the elliptic curve:
$$y^2 = x^3 - g_4 / 48 x - g_6 / 864$$ with
$$g_4 = \frac{29108241}{62500} b^{5} + \frac{81976293}{250000} b^{4} - \frac{1198505889}{125000} b^{3} - \frac{3288044907}{250000} b^{2} + \frac{5106100923}{250000} b + \frac{1196447679}{50000}$$
$$g_6 = -\frac{248712019479}{3125000} b^{5} - \frac{1355533889409}{25000000} b^{4} + \frac{10205353689741}{6250000} b^{3} + \frac{55772501101191}{25000000} b^{2} - \frac{86931059013099}{25000000} b - \frac{2533145646969}{625000}$$
Conductor norm: 1

Endomorphisms

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

Endomorphism ring over $$\Q$$:
 $$\End (J_{})$$ $$\simeq$$ $$\Z [\frac{1 + \sqrt{-3}}{2}]$$ $$\End (J_{}) \otimes \Q$$ $$\simeq$$ $$\Q(\sqrt{-3})$$ $$\End (J_{}) \otimes \R$$ $$\simeq$$ $$\C$$

Smallest field over which all endomorphisms are defined:
Galois number field $$K = \Q (a) \simeq$$ 6.6.330812181.1 with defining polynomial $$x^{6} - 21 x^{4} - 14 x^{3} + 63 x^{2} + 21 x - 35$$

Not of $$\GL_2$$-type over $$\overline{\Q}$$

Endomorphism ring over $$\overline{\Q}$$:
 $$\End (J_{\overline{\Q}})$$ $$\simeq$$ an Eichler order of index $$3$$ in a maximal order of $$\End (J_{\overline{\Q}}) \otimes \Q$$ $$\End (J_{\overline{\Q}}) \otimes \Q$$ $$\simeq$$ $$\mathrm{M}_2($$$$\Q$$$$)$$ $$\End (J_{\overline{\Q}}) \otimes \R$$ $$\simeq$$ $$\mathrm{M}_2 (\R)$$

Remainder of the endomorphism lattice by field

Over subfield $$F \simeq$$ $$\Q(\sqrt{21})$$ with generator $$\frac{9}{125} a^{5} - \frac{12}{125} a^{4} - \frac{173}{125} a^{3} + \frac{63}{125} a^{2} + \frac{483}{125} a + \frac{9}{25}$$ with minimal polynomial $$x^{2} - x - 5$$:
 $$\End (J_{F})$$ $$\simeq$$ $$\Z [\frac{1 + \sqrt{-3}}{2}]$$ $$\End (J_{F}) \otimes \Q$$ $$\simeq$$ $$\Q(\sqrt{-3})$$ $$\End (J_{F}) \otimes \R$$ $$\simeq$$ $$\C$$
Sato Tate group: $E_3$
of $$\GL_2$$-type, simple

Over subfield $$F \simeq$$ 3.3.3969.2 with generator $$-\frac{12}{125} a^{5} - \frac{9}{125} a^{4} + \frac{214}{125} a^{3} + \frac{391}{125} a^{2} - \frac{119}{125} a - \frac{112}{25}$$ with minimal polynomial $$x^{3} - 21 x - 35$$:
 $$\End (J_{F})$$ $$\simeq$$ $$\Z [\frac{1 + \sqrt{-3}}{2}]$$ $$\End (J_{F}) \otimes \Q$$ $$\simeq$$ $$\Q(\sqrt{-3})$$ $$\End (J_{F}) \otimes \R$$ $$\simeq$$ $$\C$$
Sato Tate group: $E_2$
of $$\GL_2$$-type, simple