Properties

Label 784.c.614656.1
Conductor 784
Discriminant 614656
Mordell-Weil group \(\Z/{2}\Z \times \Z/{2}\Z\)
Sato-Tate group $E_3$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\mathrm{M}_2(\R)\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathrm{M}_2(\Q)\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type yes

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Minimal equation

Minimal equation

Simplified equation

$y^2 = x^5 - 4x^4 - 13x^3 - 9x^2 - x$ (homogenize, simplify)
$y^2 = x^5z - 4x^4z^2 - 13x^3z^3 - 9x^2z^4 - xz^5$ (dehomogenize, simplify)
$y^2 = x^5 - 4x^4 - 13x^3 - 9x^2 - x$ (minimize, homogenize)

sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, -1, -9, -13, -4, 1]), R([]));
 
magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, -1, -9, -13, -4, 1], R![]);
 
sage: X = HyperellipticCurve(R([0, -1, -9, -13, -4, 1]))
 
magma: X,pi:= SimplifiedModel(C);
 

Invariants

Conductor: \( N \)  \(=\)  \(784\) \(=\) \( 2^{4} \cdot 7^{2} \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  \(=\)  \(614656\) \(=\) \( 2^{8} \cdot 7^{4} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  \(=\) \(6368\) \(=\)  \( 2^{5} \cdot 199 \)
\( I_4 \)  \(=\) \(2308096\) \(=\)  \( 2^{11} \cdot 7^{2} \cdot 23 \)
\( I_6 \)  \(=\) \(3735904256\) \(=\)  \( 2^{13} \cdot 7^{2} \cdot 41 \cdot 227 \)
\( I_{10} \)  \(=\) \(2517630976\) \(=\)  \( 2^{20} \cdot 7^{4} \)
\( J_2 \)  \(=\) \(796\) \(=\)  \( 2^{2} \cdot 199 \)
\( J_4 \)  \(=\) \(2358\) \(=\)  \( 2 \cdot 3^{2} \cdot 131 \)
\( J_6 \)  \(=\) \(-2348\) \(=\)  \( - 2^{2} \cdot 587 \)
\( J_8 \)  \(=\) \(-1857293\) \(=\)  \( -1857293 \)
\( J_{10} \)  \(=\) \(614656\) \(=\)  \( 2^{8} \cdot 7^{4} \)
\( g_1 \)  \(=\) \(1248318403996/2401\)
\( g_2 \)  \(=\) \(9291226221/4802\)
\( g_3 \)  \(=\) \(-23245787/9604\)

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

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

magma: [C![-1,0,1],C![0,0,1],C![1,0,0]];
 

Number of rational Weierstrass points: \(3\)

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

magma: MordellWeilGroupGenus2(Jacobian(C));
 

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

2-torsion field: \(\Q(\zeta_{7})^+\)

BSD invariants

Hasse-Weil conjecture: verified
Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(2\)
Regulator: \( 1 \)
Real period: \( 5.731485 \)
Tamagawa product: \( 9 \)
Torsion order:\( 12 \)
Leading coefficient: \( 0.358217 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor Cluster picture
\(2\) \(4\) \(8\) \(3\) \(1\)
\(7\) \(2\) \(4\) \(3\) \(1 + 4 T + 7 T^{2}\)

Sato-Tate group

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

Decomposition of the Jacobian

Splits over the number field \(\Q (b) \simeq \) \(\Q(\zeta_{7})^+\) with defining polynomial:
  \(x^{3} - x^{2} - 2 x + 1\)

Decomposes up to isogeny as the square of the elliptic curve:
  Elliptic curve 3.3.49.1-64.1-a3

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 \(K = \Q (a) \simeq \) \(\Q(\zeta_{7})^+\) with defining polynomial \(x^{3} - x^{2} - 2 x + 1\)

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