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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 205350.d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 205350.d1 | 205350du4 | \([1, 1, 0, -10967692875, -91021127596875]\) | \(3639478711331685826729/2016912141902025000\) | \(80856949173605953641847265625000\) | \([2]\) | \(756449280\) | \(4.8130\) | |
| 205350.d2 | 205350du2 | \([1, 1, 0, -6689567875, 209350306778125]\) | \(825824067562227826729/5613755625000000\) | \(225052516573981259765625000000\) | \([2, 2]\) | \(378224640\) | \(4.4664\) | |
| 205350.d3 | 205350du1 | \([1, 1, 0, -6678615875, 210073872562125]\) | \(821774646379511057449/38361600000\) | \(1537896409554600000000000\) | \([2]\) | \(189112320\) | \(4.1198\) | \(\Gamma_0(N)\)-optimal |
| 205350.d4 | 205350du3 | \([1, 1, 0, -2586674875, 463413750017125]\) | \(-47744008200656797609/2286529541015625000\) | \(-91665769192850589752197265625000\) | \([2]\) | \(756449280\) | \(4.8130\) |
Rank
sage: E.rank()
The elliptic curves in class 205350.d have rank \(1\).
Complex multiplication
The elliptic curves in class 205350.d do not have complex multiplication.Modular form 205350.2.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 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.
