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SageMath
E = EllipticCurve("ff1")
E.isogeny_class()
Elliptic curves in class 121275.ff
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 121275.ff1 | 121275ee4 | \([1, -1, 0, -23411817, -43124096534]\) | \(1058993490188089/13182390375\) | \(17665663874554458984375\) | \([2]\) | \(10616832\) | \(3.0786\) | |
| 121275.ff2 | 121275ee2 | \([1, -1, 0, -2739942, 679606591]\) | \(1697509118089/833765625\) | \(1117325679365478515625\) | \([2, 2]\) | \(5308416\) | \(2.7320\) | |
| 121275.ff3 | 121275ee1 | \([1, -1, 0, -2243817, 1293313216]\) | \(932288503609/779625\) | \(1044772063822265625\) | \([2]\) | \(2654208\) | \(2.3854\) | \(\Gamma_0(N)\)-optimal |
| 121275.ff4 | 121275ee3 | \([1, -1, 0, 9993933, 5200132216]\) | \(82375335041831/56396484375\) | \(-75576682857513427734375\) | \([2]\) | \(10616832\) | \(3.0786\) |
Rank
sage: E.rank()
The elliptic curves in class 121275.ff have rank \(2\).
Complex multiplication
The elliptic curves in class 121275.ff do not have complex multiplication.Modular form 121275.2.a.ff
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.
