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SageMath
E = EllipticCurve("cm1")
E.isogeny_class()
Elliptic curves in class 121680cm
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 121680.g3 | 121680cm1 | \([0, 0, 0, -673803, -215231302]\) | \(-63378025803/812500\) | \(-433717749504000000\) | \([2]\) | \(2322432\) | \(2.1936\) | \(\Gamma_0(N)\)-optimal |
| 121680.g2 | 121680cm2 | \([0, 0, 0, -10813803, -13687235302]\) | \(261984288445803/42250\) | \(22553322974208000\) | \([2]\) | \(4644864\) | \(2.5401\) | |
| 121680.g4 | 121680cm3 | \([0, 0, 0, 2368197, -1094910102]\) | \(3774555693/3515200\) | \(-1367922187690042982400\) | \([2]\) | \(6967296\) | \(2.7429\) | |
| 121680.g1 | 121680cm4 | \([0, 0, 0, -12233403, -9864631062]\) | \(520300455507/193072360\) | \(75133126158875610808320\) | \([2]\) | \(13934592\) | \(3.0894\) |
Rank
sage: E.rank()
The elliptic curves in class 121680cm have rank \(0\).
Complex multiplication
The elliptic curves in class 121680cm do not have complex multiplication.Modular form 121680.2.a.cm
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
