Show commands:
SageMath
E = EllipticCurve("m1")
E.isogeny_class()
Elliptic curves in class 4680.m
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 4680.m1 | 4680v3 | \([0, 0, 0, -14907, 697894]\) | \(490757540836/2142075\) | \(1599050419200\) | \([4]\) | \(12288\) | \(1.1946\) | |
| 4680.m2 | 4680v2 | \([0, 0, 0, -1407, -1406]\) | \(1650587344/950625\) | \(177409440000\) | \([2, 2]\) | \(6144\) | \(0.84805\) | |
| 4680.m3 | 4680v1 | \([0, 0, 0, -1002, -12179]\) | \(9538484224/26325\) | \(307054800\) | \([2]\) | \(3072\) | \(0.50148\) | \(\Gamma_0(N)\)-optimal |
| 4680.m4 | 4680v4 | \([0, 0, 0, 5613, -11234]\) | \(26198797244/15234375\) | \(-11372400000000\) | \([2]\) | \(12288\) | \(1.1946\) |
Rank
sage: E.rank()
The elliptic curves in class 4680.m have rank \(1\).
Complex multiplication
The elliptic curves in class 4680.m do not have complex multiplication.Modular form 4680.2.a.m
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.
