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SageMath
E = EllipticCurve("y1")
E.isogeny_class()
Elliptic curves in class 68450y
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 68450.bn3 | 68450y1 | \([1, 1, 1, -2567588, -866912219]\) | \(46694890801/18944000\) | \(759455017064000000000\) | \([2]\) | \(3151872\) | \(2.7044\) | \(\Gamma_0(N)\)-optimal |
| 68450.bn4 | 68450y2 | \([1, 1, 1, 8384412, -6299104219]\) | \(1625964918479/1369000000\) | \(-54882491467515625000000\) | \([2]\) | \(6303744\) | \(3.0510\) | |
| 68450.bn1 | 68450y3 | \([1, 1, 1, -180537588, -933758272219]\) | \(16232905099479601/4052240\) | \(162452174743846250000\) | \([2]\) | \(9455616\) | \(3.2537\) | |
| 68450.bn2 | 68450y4 | \([1, 1, 1, -179853088, -941189204219]\) | \(-16048965315233521/256572640900\) | \(-10285862509125055126562500\) | \([2]\) | \(18911232\) | \(3.6003\) |
Rank
sage: E.rank()
The elliptic curves in class 68450y have rank \(0\).
Complex multiplication
The elliptic curves in class 68450y do not have complex multiplication.Modular form 68450.2.a.y
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.
