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SageMath
E = EllipticCurve("m1")
E.isogeny_class()
Elliptic curves in class 2178.m
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 2178.m1 | 2178m3 | \([1, -1, 1, -10960808, -13964543161]\) | \(112763292123580561/1932612\) | \(2495906494505028\) | \([2]\) | \(96000\) | \(2.4961\) | |
| 2178.m2 | 2178m4 | \([1, -1, 1, -10949918, -13993684801]\) | \(-112427521449300721/466873642818\) | \(-602952355269793896642\) | \([2]\) | \(192000\) | \(2.8427\) | |
| 2178.m3 | 2178m1 | \([1, -1, 1, -49028, 3057959]\) | \(10091699281/2737152\) | \(3534944134284288\) | \([2]\) | \(19200\) | \(1.6914\) | \(\Gamma_0(N)\)-optimal |
| 2178.m4 | 2178m2 | \([1, -1, 1, 125212, 19784999]\) | \(168105213359/228637728\) | \(-295278302216934432\) | \([2]\) | \(38400\) | \(2.0380\) |
Rank
sage: E.rank()
The elliptic curves in class 2178.m have rank \(0\).
Complex multiplication
The elliptic curves in class 2178.m do not have complex multiplication.Modular form 2178.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
