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SageMath
E = EllipticCurve("bl1")
E.isogeny_class()
Elliptic curves in class 68450bl
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 68450.bm3 | 68450bl1 | \([1, 0, 0, -713, -86533]\) | \(-25/2\) | \(-3207158011250\) | \([]\) | \(103680\) | \(1.0794\) | \(\Gamma_0(N)\)-optimal |
| 68450.bm1 | 68450bl2 | \([1, 0, 0, -171838, -27432308]\) | \(-349938025/8\) | \(-12828632045000\) | \([]\) | \(311040\) | \(1.6287\) | |
| 68450.bm2 | 68450bl3 | \([1, 0, 0, -103388, 15417392]\) | \(-121945/32\) | \(-32071580112500000\) | \([]\) | \(518400\) | \(1.8841\) | |
| 68450.bm4 | 68450bl4 | \([1, 0, 0, 752237, -113781983]\) | \(46969655/32768\) | \(-32841298035200000000\) | \([]\) | \(1555200\) | \(2.4334\) |
Rank
sage: E.rank()
The elliptic curves in class 68450bl have rank \(1\).
Complex multiplication
The elliptic curves in class 68450bl do not have complex multiplication.Modular form 68450.2.a.bl
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 5 & 15 \\ 3 & 1 & 15 & 5 \\ 5 & 15 & 1 & 3 \\ 15 & 5 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
