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Properties

Label 48841.b.830297.1
Conductor $48841$
Discriminant $-830297$
Mordell-Weil group trivial
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

$y^2 + (x^3 + x + 1)y = -x^6 + 2x^5 - 5x^4 + 6x^3 - 6x^2 + 2x - 1$ (homogenize, simplify)
$y^2 + (x^3 + xz^2 + z^3)y = -x^6 + 2x^5z - 5x^4z^2 + 6x^3z^3 - 6x^2z^4 + 2xz^5 - z^6$ (dehomogenize, simplify)
$y^2 = -3x^6 + 8x^5 - 18x^4 + 26x^3 - 23x^2 + 10x - 3$ (homogenize, minimize)

Copy content sage:R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-1, 2, -6, 6, -5, 2, -1]), R([1, 1, 0, 1]));
 
Copy content magma:R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-1, 2, -6, 6, -5, 2, -1], R![1, 1, 0, 1]);
 
Copy content sage:X = HyperellipticCurve(R([-3, 10, -23, 26, -18, 8, -3]))
 
Copy content magma:X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(48841\) \(=\) \( 13^{2} \cdot 17^{2} \)
Copy content magma:Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-830297\) \(=\) \( - 13^{2} \cdot 17^{3} \)
Copy content magma:Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(764\) \(=\)  \( 2^{2} \cdot 191 \)
\( I_4 \)  \(=\) \(10777\) \(=\)  \( 13 \cdot 829 \)
\( I_6 \)  \(=\) \(650195\) \(=\)  \( 5 \cdot 7 \cdot 13 \cdot 1429 \)
\( I_{10} \)  \(=\) \(106278016\) \(=\)  \( 2^{7} \cdot 13^{2} \cdot 17^{3} \)
\( J_2 \)  \(=\) \(191\) \(=\)  \( 191 \)
\( J_4 \)  \(=\) \(1071\) \(=\)  \( 3^{2} \cdot 7 \cdot 17 \)
\( J_6 \)  \(=\) \(30923\) \(=\)  \( 17^{2} \cdot 107 \)
\( J_8 \)  \(=\) \(1189813\) \(=\)  \( 17^{2} \cdot 23 \cdot 179 \)
\( J_{10} \)  \(=\) \(830297\) \(=\)  \( 13^{2} \cdot 17^{3} \)
\( g_1 \)  \(=\) \(254194901951/830297\)
\( g_2 \)  \(=\) \(438975873/48841\)
\( g_3 \)  \(=\) \(3903467/2873\)

  (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

This curve has no rational points.
This curve has no rational points.
This curve has no rational points.

Copy content magma:[]; // minimal model
 
Copy content magma:[]; // simplified model
 

Number of rational Weierstrass points: \(0\)

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

This curve is locally solvable except over $\R$, $\Q_{2}$, and $\Q_{17}$.

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: trivial

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

2-torsion field: 6.0.53139008.1

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(0\)
Regulator: \( 1 \)
Real period: \( 4.401272 \)
Tamagawa product: \( 1 \)
Torsion order:\( 1 \)
Leading coefficient: \( 4.401272 \)
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?
\(13\) \(2\) \(2\) \(1\) \(1\) \(1 + 2 T + 13 T^{2}\) yes
\(17\) \(2\) \(3\) \(1\) \(1\) \(1 - T + T^{2}\) 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 \(\Q (b) \simeq \) \(\Q(\zeta_{13})^+\) with defining polynomial:
  \(x^{6} - x^{5} - 5 x^{4} + 4 x^{3} + 6 x^{2} - 3 x - 1\)

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
  \(g_4 = \frac{15771}{16} b^{5} - \frac{4229}{2} b^{4} - \frac{40285}{16} b^{3} + \frac{54719}{8} b^{2} - \frac{7461}{4} b - \frac{3507}{4}\)
  \(g_6 = \frac{6976775}{64} b^{5} - \frac{3726177}{16} b^{4} - \frac{17947943}{64} b^{3} + \frac{12075635}{16} b^{2} - \frac{6510153}{32} b - \frac{6141551}{64}\)
   Conductor norm: 24137569

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 \(K = \Q (a) \simeq \) \(\Q(\zeta_{13})^+\) with defining polynomial \(x^{6} - x^{5} - 5 x^{4} + 4 x^{3} + 6 x^{2} - 3 x - 1\)

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 \(F \simeq \) \(\Q(\sqrt{13}) \) with generator \(-a^{4} + a^{3} + 4 a^{2} - 2 a - 2\) 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 \(F \simeq \) 3.3.169.1 with generator \(a^{3} - a^{2} - 3 a + 2\) with minimal polynomial \(x^{3} - x^{2} - 4 x - 1\):

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