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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 228888d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 228888.k4 | 228888d1 | \([0, 0, 0, 1734, -761515]\) | \(2048/891\) | \(-250852678891056\) | \([2]\) | \(884736\) | \(1.4421\) | \(\Gamma_0(N)\)-optimal |
| 228888.k3 | 228888d2 | \([0, 0, 0, -115311, -14689870]\) | \(37642192/1089\) | \(4905563498313984\) | \([2, 2]\) | \(1769472\) | \(1.7887\) | |
| 228888.k2 | 228888d3 | \([0, 0, 0, -271371, 33595094]\) | \(122657188/43923\) | \(791430911061322752\) | \([2]\) | \(3538944\) | \(2.1352\) | |
| 228888.k1 | 228888d4 | \([0, 0, 0, -1831971, -954389554]\) | \(37736227588/33\) | \(594613757371392\) | \([2]\) | \(3538944\) | \(2.1352\) |
Rank
sage: E.rank()
The elliptic curves in class 228888d have rank \(1\).
Complex multiplication
The elliptic curves in class 228888d do not have complex multiplication.Modular form 228888.2.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 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.
