Properties

Label 686.a.686.1
Conductor 686
Discriminant 686
Mordell-Weil group \(\Z/{6}\Z\)
Sato-Tate group $N(G_{1,3})$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\C \times \R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\mathrm{CM} \times \Q\)
\(\overline{\Q}\)-simple no
\(\mathrm{GL}_2\)-type yes

Related objects

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Minimal equation

Minimal equation

Simplified equation

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 1, 1, 2, 1, 1], R![0, 1, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 1, 1, 2, 1, 1]), R([0, 1, 1]))
 

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 1, 1, 2, 1, 1], R![0, 1, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 4, 5, 10, 5, 4]))
 

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

Invariants

\( N \)  =  \(686\) = \( 2 \cdot 7^{3} \)
magma: Conductor(LSeries(C)); Factorization($1);
 
\( \Delta \)  =  \(686\) = \( 2 \cdot 7^{3} \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

magma: IgusaClebschInvariants(C); [Factorization(Integers()!a): a in $1];
 
sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]
 

Igusa invariants

magma: IgusaInvariants(C); [Factorization(Integers()!a): a in $1];
 

G2 invariants

magma: G2Invariants(C);
 

\( I_2 \)  = \(840\) =  \( 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
\( I_4 \)  = \(17220\) =  \( 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 41 \)
\( I_6 \)  = \(5121480\) =  \( 2^{3} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 13 \cdot 67 \)
\( I_{10} \)  = \(2809856\) =  \( 2^{13} \cdot 7^{3} \)
\( J_2 \)  = \(105\) =  \( 3 \cdot 5 \cdot 7 \)
\( J_4 \)  = \(280\) =  \( 2^{3} \cdot 5 \cdot 7 \)
\( J_6 \)  = \(-980\) =  \( - 2^{2} \cdot 5 \cdot 7^{2} \)
\( J_8 \)  = \(-45325\) =  \( - 5^{2} \cdot 7^{2} \cdot 37 \)
\( J_{10} \)  = \(686\) =  \( 2 \cdot 7^{3} \)
\( g_1 \)  = \(37209375/2\)
\( g_2 \)  = \(472500\)
\( g_3 \)  = \(-15750\)

Automorphism group

magma: AutomorphismGroup(C); IdentifyGroup($1);
 
\(\mathrm{Aut}(X)\)\(\simeq\) $C_2^2$
magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
 
\(\mathrm{Aut}(X_{\overline{\Q}})\)\(\simeq\) $C_2^2$

Rational points

magma: [C![0,0,1],C![1,0,0]];
 

Points: \((0 : 0 : 1),\, (1 : 0 : 0)\)

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

Number of rational Weierstrass points: \(2\)

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);
 

This curve is locally solvable everywhere.

Mordell-Weil group of the Jacobian:

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Group structure: \(\Z/{6}\Z\)

Generator Height Order
\(x^2 + xz + z^2\) \(=\) \(0,\) \(y\) \(=\) \(z^3\) \(0\) \(6\)

2-torsion field: 4.0.2744.1

BSD invariants

Analytic rank: \(0\)
Mordell-Weil rank: \(0\)
2-Selmer rank:\(1\)
Regulator: \( 1 \)
Real period: \( 11.49165 \)
Tamagawa product: \( 1 \)
Torsion order:\( 6 \)
Leading coefficient: \( 0.319212 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

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

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $N(G_{1,3})$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{U}(1)\times\mathrm{SU}(2)\)

Decomposition of the Jacobian

Splits over \(\Q\)

Decomposes up to isogeny as the product of the non-isogenous elliptic curves:
  Elliptic curve 14.a5
  Elliptic curve 49.a4

Endomorphisms of the Jacobian

Of \(\GL_2\)-type over \(\Q\)

Endomorphism ring over \(\Q\):

\(\End (J_{})\)\(\simeq\)an order of index \(2\) in \(\Z \times \Z\)
\(\End (J_{}) \otimes \Q \)\(\simeq\)\(\Q\) \(\times\) \(\Q\)
\(\End (J_{}) \otimes \R\)\(\simeq\) \(\R \times \R\)

Smallest field over which all endomorphisms are defined:
Galois number field \(K = \Q (a) \simeq \) \(\Q(\sqrt{-7}) \) with defining polynomial \(x^{2} - x + 2\)

Not of \(\GL_2\)-type over \(\overline{\Q}\)

Endomorphism ring over \(\overline{\Q}\):

\(\End (J_{\overline{\Q}})\)\(\simeq\)an order of index \(4\) in \(\Z \times \Z [\frac{1 + \sqrt{-7}}{2}]\)
\(\End (J_{\overline{\Q}}) \otimes \Q \)\(\simeq\)\(\Q\) \(\times\) \(\Q(\sqrt{-7}) \)
\(\End (J_{\overline{\Q}}) \otimes \R\)\(\simeq\) \(\R \times \C\)