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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 12342.d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 12342.d1 | 12342c3 | \([1, 1, 0, -234379, 40380355]\) | \(803760366578833/65593817586\) | \(116203449076471746\) | \([2]\) | \(184320\) | \(2.0172\) | |
| 12342.d2 | 12342c2 | \([1, 1, 0, -49249, -3495455]\) | \(7457162887153/1370924676\) | \(2428676689939236\) | \([2, 2]\) | \(92160\) | \(1.6706\) | |
| 12342.d3 | 12342c1 | \([1, 1, 0, -46829, -3919923]\) | \(6411014266033/296208\) | \(524750540688\) | \([2]\) | \(46080\) | \(1.3241\) | \(\Gamma_0(N)\)-optimal |
| 12342.d4 | 12342c4 | \([1, 1, 0, 97161, -20156913]\) | \(57258048889007/132611470002\) | \(-234929308408213122\) | \([2]\) | \(184320\) | \(2.0172\) |
Rank
sage: E.rank()
The elliptic curves in class 12342.d have rank \(0\).
Complex multiplication
The elliptic curves in class 12342.d do not have complex multiplication.Modular form 12342.2.a.d
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.
