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SageMath
E = EllipticCurve("bm1")
E.isogeny_class()
Elliptic curves in class 219912.bm
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 219912.bm1 | 219912c3 | \([0, 1, 0, -1985104, -58695904]\) | \(3590504967602306/2071799959977\) | \(499190156270252181504\) | \([2]\) | \(7077888\) | \(2.6612\) | |
| 219912.bm2 | 219912c2 | \([0, 1, 0, -1402984, -638487424]\) | \(2535093488117092/7367303889\) | \(887558077682648064\) | \([2, 2]\) | \(3538944\) | \(2.3146\) | |
| 219912.bm3 | 219912c1 | \([0, 1, 0, -1402004, -639425088]\) | \(10119139303540048/85833\) | \(2585130653952\) | \([2]\) | \(1769472\) | \(1.9680\) | \(\Gamma_0(N)\)-optimal |
| 219912.bm4 | 219912c4 | \([0, 1, 0, -836544, -1158252768]\) | \(-268702931670626/2248659199809\) | \(-541803532694177875968\) | \([2]\) | \(7077888\) | \(2.6612\) |
Rank
sage: E.rank()
The elliptic curves in class 219912.bm have rank \(0\).
Complex multiplication
The elliptic curves in class 219912.bm do not have complex multiplication.Modular form 219912.2.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 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.
