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SageMath
E = EllipticCurve("do1")
E.isogeny_class()
Elliptic curves in class 59850.do
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 59850.do1 | 59850ea3 | \([1, -1, 1, -3804680, 2036879947]\) | \(19804628171203875/5638671302656\) | \(1734155738284032000000\) | \([2]\) | \(3981312\) | \(2.7821\) | |
| 59850.do2 | 59850ea1 | \([1, -1, 1, -3492305, 2512853697]\) | \(11165451838341046875/572244736\) | \(241415748000000\) | \([2]\) | \(1327104\) | \(2.2328\) | \(\Gamma_0(N)\)-optimal |
| 59850.do3 | 59850ea2 | \([1, -1, 1, -3486305, 2521913697]\) | \(-11108001800138902875/79947274872976\) | \(-33727756587036750000\) | \([2]\) | \(2654208\) | \(2.5793\) | |
| 59850.do4 | 59850ea4 | \([1, -1, 1, 10019320, 13427855947]\) | \(361682234074684125/462672528510976\) | \(-142293490291899072000000\) | \([2]\) | \(7962624\) | \(3.1286\) |
Rank
sage: E.rank()
The elliptic curves in class 59850.do have rank \(1\).
Complex multiplication
The elliptic curves in class 59850.do do not have complex multiplication.Modular form 59850.2.a.do
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
