Properties

Label 12321.a.36963.1
Conductor $12321$
Discriminant $-36963$
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

Related objects

Learn more about

Show commands for: SageMath / Magma

Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^3 + x + 1)y = x^5 + 3x^4 + 4x^3 + 2x^2$ (homogenize, simplify)
$y^2 + (x^3 + xz^2 + z^3)y = x^5z + 3x^4z^2 + 4x^3z^3 + 2x^2z^4$ (dehomogenize, simplify)
$y^2 = x^6 + 4x^5 + 14x^4 + 18x^3 + 9x^2 + 2x + 1$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 0, 2, 4, 3, 1]), R([1, 1, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 0, 2, 4, 3, 1], R![1, 1, 0, 1]);
 
sage: X = HyperellipticCurve(R([1, 2, 9, 18, 14, 4, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(12321\) \(=\) \( 3^{2} \cdot 37^{2} \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-36963\) \(=\) \( - 3^{3} \cdot 37^{2} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(4\) \(=\)  \( 2^{2} \)
\( I_4 \)  \(=\) \(6697\) \(=\)  \( 37 \cdot 181 \)
\( I_6 \)  \(=\) \(85285\) \(=\)  \( 5 \cdot 37 \cdot 461 \)
\( I_{10} \)  \(=\) \(-4731264\) \(=\)  \( - 2^{7} \cdot 3^{3} \cdot 37^{2} \)
\( J_2 \)  \(=\) \(1\) \(=\)  \( 1 \)
\( J_4 \)  \(=\) \(-279\) \(=\)  \( - 3^{2} \cdot 31 \)
\( J_6 \)  \(=\) \(-1107\) \(=\)  \( - 3^{3} \cdot 41 \)
\( J_8 \)  \(=\) \(-19737\) \(=\)  \( - 3^{3} \cdot 17 \cdot 43 \)
\( J_{10} \)  \(=\) \(-36963\) \(=\)  \( - 3^{3} \cdot 37^{2} \)
\( g_1 \)  \(=\) \(-1/36963\)
\( g_2 \)  \(=\) \(31/4107\)
\( g_3 \)  \(=\) \(41/1369\)

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
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 0 : 1)\) \((-1 : 0 : 1)\) \((0 : -1 : 1)\) \((-1 : 1 : 1)\)
\((-2 : 1 : 1)\) \((1 : 2 : 1)\) \((-1 : 2 : 2)\) \((1 : -5 : 1)\) \((-1 : -5 : 2)\) \((-2 : 8 : 1)\)

magma: [C![-2,1,1],C![-2,8,1],C![-1,-5,2],C![-1,0,1],C![-1,1,1],C![-1,2,2],C![0,-1,1],C![0,0,1],C![1,-5,1],C![1,-1,0],C![1,0,0],C![1,2,1]];
 

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
\((0 : -1 : 1) - (1 : -1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.098911\) \(\infty\)
\((-1 : 0 : 1) + (0 : 0 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \(x (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0.098911\) \(\infty\)

2-torsion field: 6.0.36963.1

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(2\)
Mordell-Weil rank: \(2\)
2-Selmer rank:\(2\)
Regulator: \( 0.007337 \)
Real period: \( 18.95274 \)
Tamagawa product: \( 3 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.417205 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(3\) \(2\) \(3\) \(3\) \(1 + T + T^{2}\)
\(37\) \(2\) \(2\) \(1\) \(1 + 10 T + 37 T^{2}\)

Sato-Tate group

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

Decomposition of the Jacobian

Splits over the number field \(\Q (b) \simeq \) 6.6.69343957.1 with defining polynomial:
  \(x^{6} - x^{5} - 15 x^{4} + 28 x^{3} + 15 x^{2} - 38 x - 1\)

Decomposes up to isogeny as the square of the elliptic curve:
  \(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
  \(g_4 = \frac{27938}{11} b^{5} - \frac{102400}{11} b^{4} + \frac{41669}{11} b^{3} + \frac{90470}{11} b^{2} - \frac{45088}{11} b + \frac{110611}{11}\)
  \(g_6 = \frac{32589452}{11} b^{5} - \frac{166685962}{11} b^{4} + \frac{182285865}{11} b^{3} + \frac{203348152}{11} b^{2} - \frac{315712416}{11} b - \frac{19805878}{11}\)
   Conductor norm: 729

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 \(K = \Q (a) \simeq \) 6.6.69343957.1 with defining polynomial \(x^{6} - x^{5} - 15 x^{4} + 28 x^{3} + 15 x^{2} - 38 x - 1\)

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 \(F \simeq \) \(\Q(\sqrt{37}) \) with generator \(\frac{2}{11} a^{5} + \frac{1}{11} a^{4} - \frac{23}{11} a^{3} + \frac{27}{11} a^{2} - \frac{1}{11} a - \frac{28}{11}\) with minimal polynomial \(x^{2} - x - 9\):

\(\End (J_{F})\)\(\simeq\)\(\Z [\frac{1 + \sqrt{-3}}{2}]\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q(\sqrt{-3}) \)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\C\)
  Sato Tate group: $E_3$
  Of \(\GL_2\)-type, simple

Over subfield \(F \simeq \) 3.3.1369.1 with generator \(-\frac{4}{11} a^{5} - \frac{2}{11} a^{4} + \frac{57}{11} a^{3} - \frac{32}{11} a^{2} - \frac{108}{11} a + \frac{45}{11}\) with minimal polynomial \(x^{3} - x^{2} - 12 x - 11\):

\(\End (J_{F})\)\(\simeq\)\(\Z [\frac{1 + \sqrt{-3}}{2}]\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q(\sqrt{-3}) \)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\C\)
  Sato Tate group: $E_2$
  Of \(\GL_2\)-type, simple