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SageMath
E = EllipticCurve("ch1")
E.isogeny_class()
Elliptic curves in class 27930.ch
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 27930.ch1 | 27930cd4 | \([1, 1, 1, -2588348806, 50684255884919]\) | \(16300610738133468173382620881/2228489100\) | \(262179514125900\) | \([2]\) | \(8640000\) | \(3.5818\) | |
| 27930.ch2 | 27930cd3 | \([1, 1, 1, -161771786, 791891092103]\) | \(-3979640234041473454886161/1471455901872240\) | \(-173115315399367163760\) | \([2]\) | \(4320000\) | \(3.2352\) | |
| 27930.ch3 | 27930cd2 | \([1, 1, 1, -4309306, 2964690119]\) | \(75224183150104868881/11219310000000000\) | \(1319940602190000000000\) | \([2]\) | \(1728000\) | \(2.7771\) | |
| 27930.ch4 | 27930cd1 | \([1, 1, 1, 457414, 253379783]\) | \(89962967236397039/287450726400000\) | \(-33818290510233600000\) | \([2]\) | \(864000\) | \(2.4305\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 27930.ch have rank \(0\).
Complex multiplication
The elliptic curves in class 27930.ch do not have complex multiplication.Modular form 27930.2.a.ch
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
