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SageMath
E = EllipticCurve("fc1")
E.isogeny_class()
Elliptic curves in class 333270.fc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 333270.fc1 | 333270fc4 | \([1, -1, 1, -4005435572717, -3085479024836481259]\) | \(65853432878493908038433301506521/38511703125000000\) | \(4156112258370807328125000000\) | \([2]\) | \(5449973760\) | \(5.5248\) | |
| 333270.fc2 | 333270fc2 | \([1, -1, 1, -250341176237, -48209975225356651]\) | \(16077778198622525072705635801/388799208512064000000\) | \(41958496389968646107909184000000\) | \([2, 2]\) | \(2724986880\) | \(5.1782\) | |
| 333270.fc3 | 333270fc3 | \([1, -1, 1, -241009616237, -51969761530204651]\) | \(-14346048055032350809895395801/2509530875136386550792000\) | \(-270823962239773211905426145710152000\) | \([2]\) | \(5449973760\) | \(5.5248\) | |
| 333270.fc4 | 333270fc1 | \([1, -1, 1, -16230998957, -693943559827819]\) | \(4381924769947287308715481/608122186185572352000\) | \(65627429261948842412460736512000\) | \([4]\) | \(1362493440\) | \(4.8316\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 333270.fc have rank \(0\).
Complex multiplication
The elliptic curves in class 333270.fc do not have complex multiplication.Modular form 333270.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
