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SageMath
E = EllipticCurve("bd1")
E.isogeny_class()
Elliptic curves in class 69696.bd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 69696.bd1 | 69696cw4 | \([0, 0, 0, -10211916, 12548220784]\) | \(347873904937/395307\) | \(133831413837721239552\) | \([2]\) | \(2949120\) | \(2.7753\) | |
| 69696.bd2 | 69696cw2 | \([0, 0, 0, -802956, 86994160]\) | \(169112377/88209\) | \(29863208046433665024\) | \([2, 2]\) | \(1474560\) | \(2.4288\) | |
| 69696.bd3 | 69696cw1 | \([0, 0, 0, -454476, -116936336]\) | \(30664297/297\) | \(100549522041864192\) | \([2]\) | \(737280\) | \(2.0822\) | \(\Gamma_0(N)\)-optimal |
| 69696.bd4 | 69696cw3 | \([0, 0, 0, 3030324, 677319280]\) | \(9090072503/5845851\) | \(-1979116242350012891136\) | \([2]\) | \(2949120\) | \(2.7753\) |
Rank
sage: E.rank()
The elliptic curves in class 69696.bd have rank \(1\).
Complex multiplication
The elliptic curves in class 69696.bd do not have complex multiplication.Modular form 69696.2.a.bd
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.
