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SageMath
E = EllipticCurve("o1")
E.isogeny_class()
Elliptic curves in class 21840.o
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 21840.o1 | 21840c4 | \([0, -1, 0, -1352016, -604640784]\) | \(266912903848829942596/152163375\) | \(155815296000\) | \([2]\) | \(172032\) | \(1.9086\) | |
| 21840.o2 | 21840c2 | \([0, -1, 0, -84516, -9422784]\) | \(260798860029250384/196803140625\) | \(50381604000000\) | \([2, 2]\) | \(86016\) | \(1.5620\) | |
| 21840.o3 | 21840c3 | \([0, -1, 0, -67016, -13454784]\) | \(-32506165579682596/57814914850875\) | \(-59202472807296000\) | \([2]\) | \(172032\) | \(1.9086\) | |
| 21840.o4 | 21840c1 | \([0, -1, 0, -6391, -79034]\) | \(1804588288006144/866455078125\) | \(13863281250000\) | \([2]\) | \(43008\) | \(1.2155\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 21840.o have rank \(0\).
Complex multiplication
The elliptic curves in class 21840.o do not have complex multiplication.Modular form 21840.2.a.o
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.
