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SageMath
E = EllipticCurve("m1")
E.isogeny_class()
Elliptic curves in class 4275.m
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 4275.m1 | 4275e3 | \([1, -1, 0, -22842, -1323059]\) | \(115714886617/1539\) | \(17530171875\) | \([2]\) | \(6144\) | \(1.1090\) | |
| 4275.m2 | 4275e2 | \([1, -1, 0, -1467, -19184]\) | \(30664297/3249\) | \(37008140625\) | \([2, 2]\) | \(3072\) | \(0.76244\) | |
| 4275.m3 | 4275e1 | \([1, -1, 0, -342, 2191]\) | \(389017/57\) | \(649265625\) | \([2]\) | \(1536\) | \(0.41587\) | \(\Gamma_0(N)\)-optimal |
| 4275.m4 | 4275e4 | \([1, -1, 0, 1908, -96809]\) | \(67419143/390963\) | \(-4453312921875\) | \([2]\) | \(6144\) | \(1.1090\) |
Rank
sage: E.rank()
The elliptic curves in class 4275.m have rank \(0\).
Complex multiplication
The elliptic curves in class 4275.m do not have complex multiplication.Modular form 4275.2.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.
