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Properties

Label 37636.a.602176.1
Conductor $37636$
Discriminant $-602176$
Mordell-Weil group \(\Z \oplus \Z\)
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

$y^2 + (x^3 + x + 1)y = x^5 + 4x^4 + 6x^3 + 3x^2$ (homogenize, simplify)
$y^2 + (x^3 + xz^2 + z^3)y = x^5z + 4x^4z^2 + 6x^3z^3 + 3x^2z^4$ (dehomogenize, simplify)
$y^2 = x^6 + 4x^5 + 18x^4 + 26x^3 + 13x^2 + 2x + 1$ (homogenize, minimize)

Copy content sage:R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 0, 3, 6, 4, 1]), R([1, 1, 0, 1]));
 
Copy content magma:R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 0, 3, 6, 4, 1], R![1, 1, 0, 1]);
 
Copy content sage:X = HyperellipticCurve(R([1, 2, 13, 26, 18, 4, 1]))
 
Copy content magma:X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(37636\) \(=\) \( 2^{2} \cdot 97^{2} \)
Copy content magma:Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-602176\) \(=\) \( - 2^{6} \cdot 97^{2} \)
Copy content magma:Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(196\) \(=\)  \( 2^{2} \cdot 7^{2} \)
\( I_4 \)  \(=\) \(28033\) \(=\)  \( 17^{2} \cdot 97 \)
\( I_6 \)  \(=\) \(1517953\) \(=\)  \( 97 \cdot 15649 \)
\( I_{10} \)  \(=\) \(-77078528\) \(=\)  \( - 2^{13} \cdot 97^{2} \)
\( J_2 \)  \(=\) \(49\) \(=\)  \( 7^{2} \)
\( J_4 \)  \(=\) \(-1068\) \(=\)  \( - 2^{2} \cdot 3 \cdot 89 \)
\( J_6 \)  \(=\) \(-4912\) \(=\)  \( - 2^{4} \cdot 307 \)
\( J_8 \)  \(=\) \(-345328\) \(=\)  \( - 2^{4} \cdot 113 \cdot 191 \)
\( J_{10} \)  \(=\) \(-602176\) \(=\)  \( - 2^{6} \cdot 97^{2} \)
\( g_1 \)  \(=\) \(-282475249/602176\)
\( g_2 \)  \(=\) \(31412283/150544\)
\( g_3 \)  \(=\) \(737107/37636\)

  (Igusa-Clebsch invariants, (Igusa invariants, Igusa invariants) G2 invariants)

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

Automorphism group

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

Rational points

Known points
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 0 : 1)\) \((-1 : 0 : 1)\) \((0 : -1 : 1)\) \((-1 : 1 : 1)\)
\((-3 : 2 : 1)\) \((1 : 6 : 2)\) \((-2 : 12 : 3)\) \((-2 : -13 : 3)\) \((1 : -19 : 2)\) \((-3 : 27 : 1)\)
Known points
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 0 : 1)\) \((-1 : 0 : 1)\) \((0 : -1 : 1)\) \((-1 : 1 : 1)\)
\((-3 : 2 : 1)\) \((1 : 6 : 2)\) \((-2 : 12 : 3)\) \((-2 : -13 : 3)\) \((1 : -19 : 2)\) \((-3 : 27 : 1)\)
Known points
\((1 : -1 : 0)\) \((1 : 1 : 0)\) \((0 : -1 : 1)\) \((0 : 1 : 1)\) \((-1 : -1 : 1)\) \((-1 : 1 : 1)\)
\((1 : -25 : 2)\) \((1 : 25 : 2)\) \((-3 : -25 : 1)\) \((-3 : 25 : 1)\) \((-2 : -25 : 3)\) \((-2 : 25 : 3)\)

Copy content magma:[C![-3,2,1],C![-3,27,1],C![-2,-13,3],C![-2,12,3],C![-1,0,1],C![-1,1,1],C![0,-1,1],C![0,0,1],C![1,-19,2],C![1,-1,0],C![1,0,0],C![1,6,2]]; // minimal model
 
Copy content magma:[C![-3,-25,1],C![-3,25,1],C![-2,-25,3],C![-2,25,3],C![-1,-1,1],C![-1,1,1],C![0,-1,1],C![0,1,1],C![1,-25,2],C![1,-1,0],C![1,1,0],C![1,25,2]]; // simplified model
 

Number of rational Weierstrass points: \(0\)

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

This curve is locally solvable everywhere.

Copy content 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: \(\Z \oplus \Z\)

Copy content magma:MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\((-1 : 0 : 1) + (0 : 0 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \(x (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0.169360\) \(\infty\)
\((0 : 0 : 1) - (1 : 0 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(-x^3\) \(0.169360\) \(\infty\)
Generator $D_0$ Height Order
\((-1 : 0 : 1) + (0 : 0 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \(x (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(0\) \(0.169360\) \(\infty\)
\((0 : 0 : 1) - (1 : 0 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(-x^3\) \(0.169360\) \(\infty\)
Generator $D_0$ Height Order
\((-1 : -1 : 1) + (0 : 1 : 1) - (1 : -1 : 0) - (1 : 1 : 0)\) \(x (x + z)\) \(=\) \(0,\) \(y\) \(=\) \(x^3 + xz^2 + z^3\) \(0.169360\) \(\infty\)
\((0 : 1 : 1) - (1 : 1 : 0)\) \(z x\) \(=\) \(0,\) \(y\) \(=\) \(-x^3 + xz^2 + z^3\) \(0.169360\) \(\infty\)

2-torsion field: 6.0.602176.1

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(2\)
Mordell-Weil rank: \(2\)
2-Selmer rank:\(2\)
Regulator: \( 0.021512 \)
Real period: \( 15.40733 \)
Tamagawa product: \( 3 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.994335 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa Root number* L-factor Cluster picture Tame reduction?
\(2\) \(2\) \(6\) \(3\) \(1^*\) \(1 + T + T^{2}\) yes
\(97\) \(2\) \(2\) \(1\) \(1\) \(1 + 14 T + 97 T^{2}\) yes

Galois representations

For primes $\ell \ge 5$ the Galois representation data has not been computed for this curve since it is not generic.

For primes $\ell \le 3$, the image of the mod-$\ell$ Galois representation is listed in the table below, whenever it is not all of $\GSp(4,\F_\ell)$.

Prime \(\ell\) mod-\(\ell\) image Is torsion prime?
\(2\) 2.40.3 no
\(3\) 3.480.12 no

Sato-Tate group

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

Decomposition of the Jacobian

Splits over the number field \(\Q (b) \simeq \) 6.6.8587340257.1 with defining polynomial:
  \(x^{6} - x^{5} - 40 x^{4} - 45 x^{3} + 236 x^{2} + 230 x - 389\)

Decomposes up to isogeny as the square of the elliptic curve isogeny class:
  \(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
  \(g_4 = -\frac{780253}{376} b^{5} - \frac{3367987}{1504} b^{4} + \frac{117849811}{1504} b^{3} + \frac{385459869}{1504} b^{2} + \frac{16108321}{376} b - \frac{584419305}{1504}\)
  \(g_6 = \frac{14792680517}{6016} b^{5} + \frac{15949868983}{6016} b^{4} - \frac{558559142863}{6016} b^{3} - \frac{913242130137}{3008} b^{2} - \frac{304797991689}{6016} b + \frac{173053323651}{376}\)
   Conductor norm: 64

Copy content magma:HeuristicDecompositionFactors(C);
 

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 \(K = \Q (a) \simeq \) 6.6.8587340257.1 with defining polynomial \(x^{6} - x^{5} - 40 x^{4} - 45 x^{3} + 236 x^{2} + 230 x - 389\)

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 \(F \simeq \) \(\Q(\sqrt{97}) \) with generator \(-\frac{3}{47} a^{5} + \frac{10}{47} a^{4} + \frac{81}{47} a^{3} - \frac{7}{47} a^{2} - \frac{206}{47} a - \frac{84}{47}\) with minimal polynomial \(x^{2} - x - 24\):

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

Over subfield \(F \simeq \) 3.3.9409.1 with generator \(-\frac{9}{188} a^{5} + \frac{15}{94} a^{4} + \frac{145}{94} a^{3} - \frac{209}{188} a^{2} - \frac{1699}{188} a + \frac{453}{188}\) with minimal polynomial \(x^{3} - x^{2} - 32 x + 79\):

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

Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 

Copy content magma:HeuristicIsGL2(C); HeuristicEndomorphismDescription(C); HeuristicEndomorphismFieldOfDefinition(C);
 

Copy content magma:HeuristicIsGL2(C : Geometric := true); HeuristicEndomorphismDescription(C : Geometric := true); HeuristicEndomorphismLatticeDescription(C);