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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 1694d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 1694.c4 | 1694d1 | \([1, -1, 0, -446, 119444]\) | \(-5545233/3469312\) | \(-6146097836032\) | \([2]\) | \(2880\) | \(1.1330\) | \(\Gamma_0(N)\)-optimal |
| 1694.c3 | 1694d2 | \([1, -1, 0, -39166, 2961492]\) | \(3750606459153/45914176\) | \(81339763548736\) | \([2, 2]\) | \(5760\) | \(1.4795\) | |
| 1694.c2 | 1694d3 | \([1, -1, 0, -73046, -2899748]\) | \(24331017010833/12004097336\) | \(21265990680661496\) | \([2]\) | \(11520\) | \(1.8261\) | |
| 1694.c1 | 1694d4 | \([1, -1, 0, -624806, 190249164]\) | \(15226621995131793/2324168\) | \(4117405386248\) | \([2]\) | \(11520\) | \(1.8261\) |
Rank
sage: E.rank()
The elliptic curves in class 1694d have rank \(1\).
Complex multiplication
The elliptic curves in class 1694d do not have complex multiplication.Modular form 1694.2.a.d
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
