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Properties

Label 1686.b
Number of curves $2$
Conductor $1686$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("b1")
 
E.isogeny_class()
 

Elliptic curves in class 1686.b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1686.b1 1686c2 \([1, 0, 0, -81395, -8955759]\) \(-59637197696894581681/84095515459248\) \(-84095515459248\) \([]\) \(15600\) \(1.5754\)  
1686.b2 1686c1 \([1, 0, 0, 925, 7041]\) \(87522470053199/71599915008\) \(-71599915008\) \([5]\) \(3120\) \(0.77065\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 1686.b have rank \(0\).

Complex multiplication

The elliptic curves in class 1686.b do not have complex multiplication.

Modular form 1686.2.a.b

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{3} + q^{4} - 4 q^{5} + q^{6} + 3 q^{7} + q^{8} + q^{9} - 4 q^{10} - 3 q^{11} + q^{12} + 4 q^{13} + 3 q^{14} - 4 q^{15} + q^{16} - 2 q^{17} + q^{18} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

sage: E.isogeny_class().matrix()
 

The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.

\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)

Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with LMFDB labels.