Properties

Label 3721.a.3721.1
Conductor 3721
Discriminant -3721
Mordell-Weil group \(\Z \times \Z\)
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

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Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^3 + x + 1)y = -x^4 + x^3 + 3x^2 + x$ (homogenize, simplify)
$y^2 + (x^3 + xz^2 + z^3)y = -x^4z^2 + x^3z^3 + 3x^2z^4 + xz^5$ (dehomogenize, simplify)
$y^2 = x^6 - 2x^4 + 6x^3 + 13x^2 + 6x + 1$ (minimize, homogenize)

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]));
 
magma: X,pi:= SimplifiedModel(C);
 
sage: X = HyperellipticCurve(R([1, 6, 13, 6, -2, 0, 1]))
 

Invariants

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

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(392\) =  \( 2^{3} \cdot 7^{2} \)
\( I_4 \)  = \(26596\) =  \( 2^{2} \cdot 61 \cdot 109 \)
\( I_6 \)  = \(2436584\) =  \( 2^{3} \cdot 61 \cdot 4993 \)
\( I_{10} \)  = \(-15241216\) =  \( - 2^{12} \cdot 61^{2} \)
\( J_2 \)  = \(49\) =  \( 7^{2} \)
\( J_4 \)  = \(-177\) =  \( - 3 \cdot 59 \)
\( J_6 \)  = \(-187\) =  \( - 11 \cdot 17 \)
\( J_8 \)  = \(-10123\) =  \( - 53 \cdot 191 \)
\( J_{10} \)  = \(-3721\) =  \( - 61^{2} \)
\( g_1 \)  = \(-282475249/3721\)
\( g_2 \)  = \(20823873/3721\)
\( g_3 \)  = \(448987/3721\)

magma: IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);
 
sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]
 

Automorphism group

\(\mathrm{Aut}(X)\)\(\simeq\) $C_6$
magma: AutomorphismGroup(C); IdentifyGroup($1);
 
\(\mathrm{Aut}(X_{\overline{\Q}})\)\(\simeq\) $D_6$
magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
 

Rational points

Known points
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 0 : 1)\) \((-1 : 0 : 1)\) \((0 : -1 : 1)\) \((-1 : 1 : 1)\)
\((1 : 1 : 1)\) \((-1 : 1 : 2)\) \((-2 : 2 : 1)\) \((1 : -4 : 1)\) \((-1 : -4 : 2)\) \((-2 : 7 : 1)\)

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]];
 

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: \(\Z \times \Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((-1 : 0 : 1) + (0 : 0 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \(x (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0.098760\) \(\infty\)
\((0 : 0 : 1) - (1 : 0 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(-x^3\) \(0.098760\) \(\infty\)

2-torsion field: 6.0.238144.2

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(2\)
Mordell-Weil rank: \(2\)
2-Selmer rank:\(2\)
Regulator: \( 0.007315 \)
Real period: \( 28.08135 \)
Tamagawa product: \( 1 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.205419 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(61\) \(2\) \(2\) \(1\) \(1 + T + 61 T^{2}\)

Sato-Tate group

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

Decomposition of the Jacobian

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 the Jacobian

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