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