Show commands:
SageMath
E = EllipticCurve("bw1")
E.isogeny_class()
Elliptic curves in class 194040.bw
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 194040.bw1 | 194040ba4 | \([0, 0, 0, -2176923, -1236256378]\) | \(12990838708516/144375\) | \(12679663328640000\) | \([2]\) | \(2359296\) | \(2.2434\) | |
| 194040.bw2 | 194040ba2 | \([0, 0, 0, -139503, -18286702]\) | \(13674725584/1334025\) | \(29290022289158400\) | \([2, 2]\) | \(1179648\) | \(1.8968\) | |
| 194040.bw3 | 194040ba1 | \([0, 0, 0, -31458, 1831277]\) | \(2508888064/396165\) | \(543640565215440\) | \([2]\) | \(589824\) | \(1.5502\) | \(\Gamma_0(N)\)-optimal |
| 194040.bw4 | 194040ba3 | \([0, 0, 0, 169197, -87867682]\) | \(6099383804/41507235\) | \(-3645352488330685440\) | \([2]\) | \(2359296\) | \(2.2434\) |
Rank
sage: E.rank()
The elliptic curves in class 194040.bw have rank \(1\).
Complex multiplication
The elliptic curves in class 194040.bw do not have complex multiplication.Modular form 194040.2.a.bw
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
