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SageMath
E = EllipticCurve("bm1")
E.isogeny_class()
Elliptic curves in class 14784bm
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 14784.cj4 | 14784bm1 | \([0, 1, 0, -2177, 38367]\) | \(4354703137/1617\) | \(423886848\) | \([2]\) | \(10240\) | \(0.62279\) | \(\Gamma_0(N)\)-optimal |
| 14784.cj3 | 14784bm2 | \([0, 1, 0, -2497, 26015]\) | \(6570725617/2614689\) | \(685425033216\) | \([2, 2]\) | \(20480\) | \(0.96936\) | |
| 14784.cj2 | 14784bm3 | \([0, 1, 0, -18177, -930465]\) | \(2533811507137/58110129\) | \(15233221656576\) | \([2, 2]\) | \(40960\) | \(1.3159\) | |
| 14784.cj6 | 14784bm4 | \([0, 1, 0, 8063, 197087]\) | \(221115865823/190238433\) | \(-49869863780352\) | \([2]\) | \(40960\) | \(1.3159\) | |
| 14784.cj1 | 14784bm5 | \([0, 1, 0, -289217, -59962977]\) | \(10206027697760497/5557167\) | \(1456777986048\) | \([2]\) | \(81920\) | \(1.6625\) | |
| 14784.cj5 | 14784bm6 | \([0, 1, 0, 1983, -2861793]\) | \(3288008303/13504609503\) | \(-3540152353554432\) | \([2]\) | \(81920\) | \(1.6625\) |
Rank
sage: E.rank()
The elliptic curves in class 14784bm have rank \(0\).
Complex multiplication
The elliptic curves in class 14784bm do not have complex multiplication.Modular form 14784.2.a.bm
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
