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SageMath
E = EllipticCurve("fd1")
E.isogeny_class()
Elliptic curves in class 394944.fd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 394944.fd1 | 394944fd3 | \([0, 1, 0, -15000289, 20719742591]\) | \(803760366578833/65593817586\) | \(30462036954702609383424\) | \([2]\) | \(35389440\) | \(3.0569\) | |
| 394944.fd2 | 394944fd2 | \([0, 1, 0, -3151969, -1780217089]\) | \(7457162887153/1370924676\) | \(636663022207431081984\) | \([2, 2]\) | \(17694720\) | \(2.7104\) | |
| 394944.fd3 | 394944fd1 | \([0, 1, 0, -2997089, -1998009345]\) | \(6411014266033/296208\) | \(137560205738115072\) | \([2]\) | \(8847360\) | \(2.3638\) | \(\Gamma_0(N)\)-optimal |
| 394944.fd4 | 394944fd4 | \([0, 1, 0, 6218271, -10338994305]\) | \(57258048889007/132611470002\) | \(-61585308623362620653568\) | \([2]\) | \(35389440\) | \(3.0569\) |
Rank
sage: E.rank()
The elliptic curves in class 394944.fd have rank \(1\).
Complex multiplication
The elliptic curves in class 394944.fd do not have complex multiplication.Modular form 394944.2.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 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.
