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SageMath
E = EllipticCurve("bf1")
E.isogeny_class()
Elliptic curves in class 138600bf
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 138600.di4 | 138600bf1 | \([0, 0, 0, 81825, -8630750]\) | \(20777545136/23059575\) | \(-67241720700000000\) | \([2]\) | \(786432\) | \(1.9154\) | \(\Gamma_0(N)\)-optimal |
| 138600.di3 | 138600bf2 | \([0, 0, 0, -462675, -81049250]\) | \(939083699236/300155625\) | \(3501015210000000000\) | \([2, 2]\) | \(1572864\) | \(2.2620\) | |
| 138600.di2 | 138600bf3 | \([0, 0, 0, -2937675, 1876675750]\) | \(120186986927618/4332064275\) | \(101058395407200000000\) | \([2]\) | \(3145728\) | \(2.6086\) | |
| 138600.di1 | 138600bf4 | \([0, 0, 0, -6699675, -6673558250]\) | \(1425631925916578/270703125\) | \(6314962500000000000\) | \([2]\) | \(3145728\) | \(2.6086\) |
Rank
sage: E.rank()
The elliptic curves in class 138600bf have rank \(0\).
Complex multiplication
The elliptic curves in class 138600bf do not have complex multiplication.Modular form 138600.2.a.bf
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
