Properties

Label 1386.b
Number of curves $4$
Conductor $1386$
CM no
Rank $1$
Graph

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

Elliptic curves in class 1386.b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1386.b1 1386c4 \([1, -1, 0, -46473, 3867749]\) \(15226621995131793/2324168\) \(1694318472\) \([2]\) \(3072\) \(1.1765\)  
1386.b2 1386c3 \([1, -1, 0, -5433, -57835]\) \(24331017010833/12004097336\) \(8750986957944\) \([2]\) \(3072\) \(1.1765\)  
1386.b3 1386c2 \([1, -1, 0, -2913, 60605]\) \(3750606459153/45914176\) \(33471434304\) \([2, 2]\) \(1536\) \(0.82991\)  
1386.b4 1386c1 \([1, -1, 0, -33, 2429]\) \(-5545233/3469312\) \(-2529128448\) \([2]\) \(768\) \(0.48333\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 1386.b have rank \(1\).

Complex multiplication

The elliptic curves in class 1386.b do not have complex multiplication.

Modular form 1386.2.a.b

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{4} - 2 q^{5} - q^{7} - q^{8} + 2 q^{10} + q^{11} + 2 q^{13} + q^{14} + q^{16} - 2 q^{17} + O(q^{20})\)  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.

\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.