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SageMath
E = EllipticCurve("fc1")
E.isogeny_class()
Elliptic curves in class 22050.fc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 22050.fc1 | 22050fq2 | \([1, -1, 1, -55355, -4999053]\) | \(-349938025/8\) | \(-428830605000\) | \([]\) | \(64800\) | \(1.3455\) | |
| 22050.fc2 | 22050fq3 | \([1, -1, 1, -33305, 2828697]\) | \(-121945/32\) | \(-1072076512500000\) | \([]\) | \(108000\) | \(1.6009\) | |
| 22050.fc3 | 22050fq1 | \([1, -1, 1, -230, -15753]\) | \(-25/2\) | \(-107207651250\) | \([]\) | \(21600\) | \(0.79616\) | \(\Gamma_0(N)\)-optimal |
| 22050.fc4 | 22050fq4 | \([1, -1, 1, 242320, -20875053]\) | \(46969655/32768\) | \(-1097806348800000000\) | \([]\) | \(324000\) | \(2.1502\) |
Rank
sage: E.rank()
The elliptic curves in class 22050.fc have rank \(1\).
Complex multiplication
The elliptic curves in class 22050.fc do not have complex multiplication.Modular form 22050.2.a.fc
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.
\(\left(\begin{array}{rrrr} 1 & 15 & 3 & 5 \\ 15 & 1 & 5 & 3 \\ 3 & 5 & 1 & 15 \\ 5 & 3 & 15 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
