Properties

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

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Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^3 + x + 1)y = -2x^5 + 3x^4 - 3x^2$ (homogenize, simplify)
$y^2 + (x^3 + xz^2 + z^3)y = -2x^5z + 3x^4z^2 - 3x^2z^4$ (dehomogenize, simplify)
$y^2 = x^6 - 8x^5 + 14x^4 + 2x^3 - 11x^2 + 2x + 1$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 0, -3, 0, 3, -2]), R([1, 1, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 0, -3, 0, 3, -2], R![1, 1, 0, 1]);
 
sage: X = HyperellipticCurve(R([1, 2, -11, 2, 14, -8, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

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

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(268\) \(=\)  \( 2^{2} \cdot 67 \)
\( I_4 \)  \(=\) \(2961\) \(=\)  \( 3^{2} \cdot 7 \cdot 47 \)
\( I_6 \)  \(=\) \(216951\) \(=\)  \( 3 \cdot 7 \cdot 10331 \)
\( I_{10} \)  \(=\) \(18816\) \(=\)  \( 2^{7} \cdot 3 \cdot 7^{2} \)
\( J_2 \)  \(=\) \(201\) \(=\)  \( 3 \cdot 67 \)
\( J_4 \)  \(=\) \(573\) \(=\)  \( 3 \cdot 191 \)
\( J_6 \)  \(=\) \(-563\) \(=\)  \( -563 \)
\( J_8 \)  \(=\) \(-110373\) \(=\)  \( - 3 \cdot 36791 \)
\( J_{10} \)  \(=\) \(35721\) \(=\)  \( 3^{6} \cdot 7^{2} \)
\( g_1 \)  \(=\) \(1350125107/147\)
\( g_2 \)  \(=\) \(57445733/441\)
\( g_3 \)  \(=\) \(-2527307/3969\)

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

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
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 0 : 1)\) \((0 : -1 : 1)\) \((-1 : -1 : 1)\) \((1 : -1 : 1)\)
\((-1 : 2 : 1)\) \((1 : -2 : 1)\) \((2 : -4 : 1)\) \((1 : -5 : 2)\) \((2 : -7 : 1)\) \((1 : -8 : 2)\)
\((1 : -16 : 6)\) \((-5 : -46 : 1)\) \((6 : -135 : 5)\) \((-5 : 175 : 1)\) \((1 : -237 : 6)\) \((6 : -356 : 5)\)

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]];
 

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
\((-1 : 2 : 1) + (1 : -2 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \((x - z) (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(-2xz^2\) \(0.064863\) \(\infty\)
\((0 : -1 : 1) + (2 : -7 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \((-x + 2z) x\) \(=\) \(0,\) \(y\) \(=\) \(-3xz^2 - z^3\) \(0.064863\) \(\infty\)

2-torsion field: 9.9.62523502209.1

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(2\)
Mordell-Weil rank: \(2\)
2-Selmer rank:\(2\)
Regulator: \( 0.003155 \)
Real period: \( 23.23416 \)
Tamagawa product: \( 3 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.219944 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(3\) \(4\) \(6\) \(3\) \(1 + 3 T + 3 T^{2}\)
\(7\) \(2\) \(2\) \(1\) \(1 + 5 T + 7 T^{2}\)

Sato-Tate group

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

Decomposition of the Jacobian

Splits over the number field \(\Q (b) \simeq \) 6.6.330812181.1 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 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 \(K = \Q (a) \simeq \) 6.6.330812181.1 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