Properties

Label 32761.a.32761.1
Conductor $32761$
Discriminant $-32761$
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

$y^2 + (x^3 + x + 1)y = x^5 + 10x^2 + 19x + 10$ (homogenize, simplify)
$y^2 + (x^3 + xz^2 + z^3)y = x^5z + 10x^2z^4 + 19xz^5 + 10z^6$ (dehomogenize, simplify)
$y^2 = x^6 + 4x^5 + 2x^4 + 2x^3 + 41x^2 + 78x + 41$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([10, 19, 10, 0, 0, 1]), R([1, 1, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![10, 19, 10, 0, 0, 1], R![1, 1, 0, 1]);
 
sage: X = HyperellipticCurve(R([41, 78, 41, 2, 2, 4, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(32761\) \(=\) \( 181^{2} \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-32761\) \(=\) \( - 181^{2} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(676\) \(=\)  \( 2^{2} \cdot 13^{2} \)
\( I_4 \)  \(=\) \(41449\) \(=\)  \( 181 \cdot 229 \)
\( I_6 \)  \(=\) \(6681253\) \(=\)  \( 181 \cdot 36913 \)
\( I_{10} \)  \(=\) \(-4193408\) \(=\)  \( - 2^{7} \cdot 181^{2} \)
\( J_2 \)  \(=\) \(169\) \(=\)  \( 13^{2} \)
\( J_4 \)  \(=\) \(-537\) \(=\)  \( - 3 \cdot 179 \)
\( J_6 \)  \(=\) \(-547\) \(=\)  \( -547 \)
\( J_8 \)  \(=\) \(-95203\) \(=\)  \( -95203 \)
\( J_{10} \)  \(=\) \(-32761\) \(=\)  \( - 181^{2} \)
\( g_1 \)  \(=\) \(-137858491849/32761\)
\( g_2 \)  \(=\) \(2591996433/32761\)
\( g_3 \)  \(=\) \(15622867/32761\)

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
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((-1 : 0 : 1)\) \((-1 : 1 : 1)\) \((-2 : 4 : 1)\) \((1 : 5 : 1)\)
\((-2 : 5 : 1)\) \((1 : -8 : 1)\) \((-4 : 30 : 3)\) \((-4 : 43 : 3)\) \((-5 : 62 : 2)\) \((-5 : 75 : 2)\)

magma: [C![-5,62,2],C![-5,75,2],C![-4,30,3],C![-4,43,3],C![-2,4,1],C![-2,5,1],C![-1,0,1],C![-1,1,1],C![1,-8,1],C![1,-1,0],C![1,0,0],C![1,5,1]];
 

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
\((-2 : 5 : 1) - (1 : -1 : 0)\) \(z (x + 2z)\) \(=\) \(0,\) \(y\) \(=\) \(5z^3\) \(0.167303\) \(\infty\)
\((-2 : 5 : 1) + (-1 : 0 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \((x + z) (x + 2z)\) \(=\) \(0,\) \(y\) \(=\) \(-5xz^2 - 5z^3\) \(0.167303\) \(\infty\)

2-torsion field: 6.0.2096704.1

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(2\)
Mordell-Weil rank: \(2\)
2-Selmer rank:\(2\)
Regulator: \( 0.020992 \)
Real period: \( 24.53325 \)
Tamagawa product: \( 1 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.515024 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(181\) \(2\) \(2\) \(1\) \(1 - 7 T + 181 T^{2}\)

Sato-Tate group

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

Decomposition of the Jacobian

Splits over the number field \(\Q (b) \simeq \) 6.6.194264244901.1 with defining polynomial:
  \(x^{6} - x^{5} - 75 x^{4} - 104 x^{3} + 918 x^{2} + 2509 x + 1685\)

Decomposes up to isogeny as the square of the elliptic curve:
  \(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
  \(g_4 = -\frac{798799}{209375} b^{5} - \frac{3129874}{209375} b^{4} + \frac{47535977}{209375} b^{3} + \frac{12936296}{8375} b^{2} + \frac{660966068}{209375} b + \frac{84325344}{41875}\)
  \(g_6 = \frac{10121301533}{5234375} b^{5} + \frac{2789901058}{5234375} b^{4} - \frac{662632165209}{5234375} b^{3} - \frac{69969753182}{209375} b^{2} + \frac{1174801020694}{5234375} b + \frac{604527106827}{1046875}\)
   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 \(K = \Q (a) \simeq \) 6.6.194264244901.1 with defining polynomial \(x^{6} - x^{5} - 75 x^{4} - 104 x^{3} + 918 x^{2} + 2509 x + 1685\)

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 \(F \simeq \) \(\Q(\sqrt{181}) \) with generator \(-\frac{4}{1675} a^{5} + \frac{13}{1675} a^{4} + \frac{187}{1675} a^{3} + \frac{414}{1675} a^{2} + \frac{254}{1675} a - \frac{2088}{335}\) with minimal polynomial \(x^{2} - x - 45\):

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

Over subfield \(F \simeq \) 3.3.32761.1 with generator \(-\frac{16}{335} a^{5} + \frac{52}{335} a^{4} + \frac{1083}{335} a^{3} - \frac{689}{335} a^{2} - \frac{13389}{335} a - \frac{2724}{67}\) with minimal polynomial \(x^{3} - x^{2} - 60 x + 67\):

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