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Properties

Label 8281.c.405769.1
Conductor $8281$
Discriminant $405769$
Mordell-Weil group \(\Z/{2}\Z \oplus \Z/{2}\Z\)
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

$y^2 + (x^2 + x)y = x^5 + 8x^4 + 11x^3 + 3x^2 - x$ (homogenize, simplify)
$y^2 + (x^2z + xz^2)y = x^5z + 8x^4z^2 + 11x^3z^3 + 3x^2z^4 - xz^5$ (dehomogenize, simplify)
$y^2 = 4x^5 + 33x^4 + 46x^3 + 13x^2 - 4x$ (homogenize, minimize)

Copy content sage:R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -1, 3, 11, 8, 1]), R([0, 1, 1]));
 
Copy content magma:R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -1, 3, 11, 8, 1], R![0, 1, 1]);
 
Copy content sage:X = HyperellipticCurve(R([0, -4, 13, 46, 33, 4]))
 
Copy content magma:X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(8281\) \(=\) \( 7^{2} \cdot 13^{2} \)
Copy content magma:Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(405769\) \(=\) \( 7^{4} \cdot 13^{2} \)
Copy content magma:Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(2596\) \(=\)  \( 2^{2} \cdot 11 \cdot 59 \)
\( I_4 \)  \(=\) \(375193\) \(=\)  \( 7^{2} \cdot 13 \cdot 19 \cdot 31 \)
\( I_6 \)  \(=\) \(248614093\) \(=\)  \( 7^{2} \cdot 13 \cdot 390289 \)
\( I_{10} \)  \(=\) \(51938432\) \(=\)  \( 2^{7} \cdot 7^{4} \cdot 13^{2} \)
\( J_2 \)  \(=\) \(649\) \(=\)  \( 11 \cdot 59 \)
\( J_4 \)  \(=\) \(1917\) \(=\)  \( 3^{3} \cdot 71 \)
\( J_6 \)  \(=\) \(-1907\) \(=\)  \( -1907 \)
\( J_8 \)  \(=\) \(-1228133\) \(=\)  \( -1228133 \)
\( J_{10} \)  \(=\) \(405769\) \(=\)  \( 7^{4} \cdot 13^{2} \)
\( g_1 \)  \(=\) \(115139273278249/405769\)
\( g_2 \)  \(=\) \(524030063733/405769\)
\( g_3 \)  \(=\) \(-803230307/405769\)

  (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

All points: \((1 : 0 : 0),\, (0 : 0 : 1),\, (-1 : 0 : 1)\)
All points: \((1 : 0 : 0),\, (0 : 0 : 1),\, (-1 : 0 : 1)\)
All points: \((1 : 0 : 0),\, (0 : 0 : 1),\, (-1 : 0 : 1)\)

Copy content magma:[C![-1,0,1],C![0,0,1],C![1,0,0]]; // minimal model
 
Copy content magma:[C![-1,0,1],C![0,0,1],C![1,0,0]]; // simplified model
 

Number of rational Weierstrass points: \(3\)

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: \(\Z/{2}\Z \oplus \Z/{2}\Z\)

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

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

2-torsion field: 3.3.8281.1

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(2\)
Regulator: \( 1 \)
Real period: \( 19.78540 \)
Tamagawa product: \( 1 \)
Torsion order:\( 4 \)
Leading coefficient: \( 1.236587 \)
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?
\(7\) \(2\) \(4\) \(1\) \(1\) \(1 + 4 T + 7 T^{2}\) yes
\(13\) \(2\) \(2\) \(1\) \(1\) \(1 + 2 T + 13 T^{2}\) 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?
\(2\) 2.240.1 yes
\(3\) 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 \(\Q (b) \simeq \) 6.6.891474493.1 with defining polynomial:
  \(x^{6} - x^{5} - 31 x^{4} + 4 x^{3} + 253 x^{2} + 101 x - 391\)

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
  \(g_4 = \frac{6900061}{184} b^{5} + \frac{2308663}{368} b^{4} - \frac{213373265}{184} b^{3} - \frac{110121753}{92} b^{2} + \frac{189026334}{23} b + \frac{212819299}{16}\)
  \(g_6 = -\frac{14940873857}{1472} b^{5} - \frac{1373744645}{736} b^{4} + \frac{7227770823}{23} b^{3} + \frac{241586544017}{736} b^{2} - \frac{3281620049799}{1472} b - \frac{14496977859}{4}\)
   Conductor norm: 1

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 \(K = \Q (a) \simeq \) 6.6.891474493.1 with defining polynomial \(x^{6} - x^{5} - 31 x^{4} + 4 x^{3} + 253 x^{2} + 101 x - 391\)

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 \(F \simeq \) \(\Q(\sqrt{13}) \) with generator \(-\frac{1}{368} a^{5} - \frac{3}{92} a^{4} + \frac{59}{368} a^{3} + \frac{211}{368} a^{2} - \frac{227}{184} a - \frac{13}{16}\) with minimal polynomial \(x^{2} - x - 3\):

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

Over subfield \(F \simeq \) 3.3.8281.1 with generator \(\frac{3}{184} a^{5} - \frac{5}{92} a^{4} - \frac{39}{184} a^{3} + \frac{195}{184} a^{2} - \frac{31}{23} a - \frac{43}{8}\) with minimal polynomial \(x^{3} - x^{2} - 30 x + 64\):

\(\End (J_{F})\)\(\simeq\)\(\Z [\frac{1 + \sqrt{-3}}{2}]\)
\(\End (J_{F}) \otimes \Q \)\(\simeq\)\(\Q(\sqrt{-3}) \)
\(\End (J_{F}) \otimes \R\)\(\simeq\) \(\C\)
  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);