Show commands:
SageMath
E = EllipticCurve("fd1")
E.isogeny_class()
Elliptic curves in class 76050fd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 76050.dk2 | 76050fd1 | \([1, -1, 1, -1236605, -525846603]\) | \(3803721481/26000\) | \(1429489652906250000\) | \([2]\) | \(2322432\) | \(2.3176\) | \(\Gamma_0(N)\)-optimal |
| 76050.dk3 | 76050fd2 | \([1, -1, 1, -476105, -1166187603]\) | \(-217081801/10562500\) | \(-580730171493164062500\) | \([2]\) | \(4644864\) | \(2.6642\) | |
| 76050.dk1 | 76050fd3 | \([1, -1, 1, -7890980, 8191384647]\) | \(988345570681/44994560\) | \(2473817613733440000000\) | \([2]\) | \(6967296\) | \(2.8670\) | |
| 76050.dk4 | 76050fd4 | \([1, -1, 1, 4277020, 31164568647]\) | \(157376536199/7722894400\) | \(-424607601357216225000000\) | \([2]\) | \(13934592\) | \(3.2135\) |
Rank
sage: E.rank()
The elliptic curves in class 76050fd have rank \(1\).
Complex multiplication
The elliptic curves in class 76050fd do not have complex multiplication.Modular form 76050.2.a.fd
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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
