Properties

Label 3721.a.3721.1
Conductor 3721
Discriminant -3721
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

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Minimal equation

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

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

Invariants

magma: Conductor(LSeries(C)); Factorization($1);
 
\( N \)  =  \( 3721 \)  =  \( 61^{2} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 
\( \Delta \)  =  \(-3721\)  =  \( -1 \cdot 61^{2} \)

Igusa-Clebsch invariants

magma: IgusaClebschInvariants(C); [Factorization(Integers()!a): a in $1];
 
sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]
 

Igusa invariants

magma: IgusaInvariants(C); [Factorization(Integers()!a): a in $1];
 

G2 invariants

magma: G2Invariants(C);
 

\( I_2 \)  =  \(392\)  =  \( 2^{3} \cdot 7^{2} \)
\( I_4 \)  =  \(26596\)  =  \( 2^{2} \cdot 61 \cdot 109 \)
\( I_6 \)  =  \(2436584\)  =  \( 2^{3} \cdot 61 \cdot 4993 \)
\( I_{10} \)  =  \(-15241216\)  =  \( -1 \cdot 2^{12} \cdot 61^{2} \)
\( J_2 \)  =  \(49\)  =  \( 7^{2} \)
\( J_4 \)  =  \(-177\)  =  \( -1 \cdot 3 \cdot 59 \)
\( J_6 \)  =  \(-187\)  =  \( -1 \cdot 11 \cdot 17 \)
\( J_8 \)  =  \(-10123\)  =  \( -1 \cdot 53 \cdot 191 \)
\( J_{10} \)  =  \(-3721\)  =  \( -1 \cdot 61^{2} \)
\( g_1 \)  =  \(-282475249/3721\)
\( g_2 \)  =  \(20823873/3721\)
\( g_3 \)  =  \(448987/3721\)
Alternative geometric invariants: Igusa-Clebsch, Igusa, 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![-2,2,1],C![-2,7,1],C![-1,-4,2],C![-1,0,1],C![-1,1,1],C![-1,1,2],C![0,-1,1],C![0,0,1],C![1,-4,1],C![1,-1,0],C![1,0,0],C![1,1,1]];
 

Known rational points: (-2 : 2 : 1), (-2 : 7 : 1), (-1 : -4 : 2), (-1 : 0 : 1), (-1 : 1 : 1), (-1 : 1 : 2), (0 : -1 : 1), (0 : 0 : 1), (1 : -4 : 1), (1 : -1 : 0), (1 : 0 : 0), (1 : 1 : 1)

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: 0.00731515919878

Real period: 28.081352476680556443659748756

Tamagawa numbers: 1 (p = 61)

magma: TorsionSubgroup(Jacobian(SimplifiedModel(C))); AbelianInvariants($1);
 

Torsion: \(\mathrm{trivial}\)

2-torsion field: 6.0.238144.2

Sato-Tate group

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

Decomposition

Splits over the number field \(\Q (b) \simeq \) 6.6.844596301.1 with defining polynomial:
  \(x^{6} - x^{5} - 25 x^{4} - 8 x^{3} + 123 x^{2} + 126 x + 27\)

Decomposes up to isogeny as the square of the elliptic curve:
  \(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
\(g_4 = \frac{6632447}{648} b^{5} - \frac{2121977}{162} b^{4} - \frac{325981945}{1296} b^{3} - \frac{3529355}{324} b^{2} + \frac{22472053}{18} b + \frac{130538657}{144}\)
\(g_6 = -\frac{2420355499}{324} b^{5} + \frac{25668283757}{2592} b^{4} + \frac{953942534383}{5184} b^{3} + \frac{1335649351}{2592} b^{2} - \frac{66892621411}{72} b - \frac{191864719165}{288}\)
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 \(K = \Q (a) \simeq \) 6.6.844596301.1 with defining polynomial \(x^{6} - x^{5} - 25 x^{4} - 8 x^{3} + 123 x^{2} + 126 x + 27\)

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 \(F \simeq \) \(\Q(\sqrt{61}) \) with generator \(-\frac{2}{81} a^{5} + \frac{5}{81} a^{4} + \frac{29}{81} a^{3} + \frac{13}{81} a^{2} - \frac{1}{9} a - \frac{31}{9}\) with minimal polynomial \(x^{2} - x - 15\):
\(\End (J_{F})\)\(\simeq\)\(\Z [\frac{1 + \sqrt{-3}}{2}]\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q(\sqrt{-3}) \)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\C\)
Sato Tate group: $E_3$
of \(\GL_2\)-type, simple

Over subfield \(F \simeq \) 3.3.3721.1 with generator \(-\frac{4}{27} a^{5} + \frac{10}{27} a^{4} + \frac{85}{27} a^{3} - \frac{82}{27} a^{2} - \frac{44}{3} a - \frac{11}{3}\) with minimal polynomial \(x^{3} - x^{2} - 20 x + 9\):
\(\End (J_{F})\)\(\simeq\)\(\Z [\frac{1 + \sqrt{-3}}{2}]\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q(\sqrt{-3}) \)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\C\)
Sato Tate group: $E_2$
of \(\GL_2\)-type, simple