Properties

Label 15876.b.222264.1
Conductor $15876$
Discriminant $-222264$
Mordell-Weil group \(\Z/{3}\Z\)
Sato-Tate group $E_3$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\mathrm{M}_2(\R)\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathrm{M}_2(\Q)\)
\(\End(J) \otimes \Q\) \(\mathsf{CM}\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type yes

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Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^3 + x + 1)y = -x^6 + 2x^5 - 4x^4 + 4x^3 - 5x^2 + 2x - 1$ (homogenize, simplify)
$y^2 + (x^3 + xz^2 + z^3)y = -x^6 + 2x^5z - 4x^4z^2 + 4x^3z^3 - 5x^2z^4 + 2xz^5 - z^6$ (dehomogenize, simplify)
$y^2 = -3x^6 + 8x^5 - 14x^4 + 18x^3 - 19x^2 + 10x - 3$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([-1, 2, -5, 4, -4, 2, -1]), R([1, 1, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![-1, 2, -5, 4, -4, 2, -1], R![1, 1, 0, 1]);
 
sage: X = HyperellipticCurve(R([-3, 10, -19, 18, -14, 8, -3]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(15876\) \(=\) \( 2^{2} \cdot 3^{4} \cdot 7^{2} \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(-222264\) \(=\) \( - 2^{3} \cdot 3^{4} \cdot 7^{3} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(636\) \(=\)  \( 2^{2} \cdot 3 \cdot 53 \)
\( I_4 \)  \(=\) \(6129\) \(=\)  \( 3^{3} \cdot 227 \)
\( I_6 \)  \(=\) \(310743\) \(=\)  \( 3^{3} \cdot 17 \cdot 677 \)
\( I_{10} \)  \(=\) \(28449792\) \(=\)  \( 2^{10} \cdot 3^{4} \cdot 7^{3} \)
\( J_2 \)  \(=\) \(159\) \(=\)  \( 3 \cdot 53 \)
\( J_4 \)  \(=\) \(798\) \(=\)  \( 2 \cdot 3 \cdot 7 \cdot 19 \)
\( J_6 \)  \(=\) \(16268\) \(=\)  \( 2^{2} \cdot 7^{2} \cdot 83 \)
\( J_8 \)  \(=\) \(487452\) \(=\)  \( 2^{2} \cdot 3 \cdot 7^{2} \cdot 829 \)
\( J_{10} \)  \(=\) \(222264\) \(=\)  \( 2^{3} \cdot 3^{4} \cdot 7^{3} \)
\( g_1 \)  \(=\) \(1254586479/2744\)
\( g_2 \)  \(=\) \(2828663/196\)
\( g_3 \)  \(=\) \(233147/126\)

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

This curve has no rational points.

magma: [];
 

Number of rational Weierstrass points: \(0\)

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

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

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: \(\Z/{3}\Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

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

2-torsion field: 6.0.14224896.1

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(0\)
Regulator: \( 1 \)
Real period: \( 5.070237 \)
Tamagawa product: \( 3 \)
Torsion order:\( 3 \)
Leading coefficient: \( 1.690079 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(2\) \(3\) \(1\) \(1 - T + T^{2}\)
\(3\) \(4\) \(4\) \(1\) \(1 + 3 T^{2}\)
\(7\) \(2\) \(3\) \(3\) \(1 + T + T^{2}\)

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $E_3$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{SU}(2)\)

Decomposition of the Jacobian

Splits over the number field \(\Q (b) \simeq \) \(\Q(\zeta_{9})^+\) with defining polynomial:
  \(x^{3} - 3 x - 1\)

Decomposes up to isogeny as the square of the elliptic curve:
  \(y^2 = x^3 - g_4 / 48 x - g_6 / 864\) with
  \(g_4 = \frac{41553}{16} b^{2} + \frac{78489}{16} b + \frac{23085}{16}\)
  \(g_6 = \frac{29032425}{64} b^{2} + \frac{54580959}{64} b + \frac{1935495}{8}\)
   Conductor norm: 2744

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 \(K = \Q (a) \simeq \) \(\Q(\zeta_{9})^+\) with defining polynomial \(x^{3} - 3 x - 1\)

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 \(\End (J_{\overline{\Q}}) \otimes \Q\)
\(\End (J_{\overline{\Q}}) \otimes \Q \)\(\simeq\)\(\mathrm{M}_2(\)\(\Q\)\()\)
\(\End (J_{\overline{\Q}}) \otimes \R\)\(\simeq\) \(\mathrm{M}_2 (\R)\)