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SageMath
E = EllipticCurve("w1")
E.isogeny_class()
Elliptic curves in class 21840w
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 21840.c3 | 21840w1 | \([0, -1, 0, -479136, 127814400]\) | \(2969894891179808929/22997520\) | \(94197841920\) | \([2]\) | \(122880\) | \(1.6978\) | \(\Gamma_0(N)\)-optimal |
| 21840.c2 | 21840w2 | \([0, -1, 0, -479456, 127635456]\) | \(2975849362756797409/8263842596100\) | \(33848699273625600\) | \([2, 2]\) | \(245760\) | \(2.0444\) | |
| 21840.c4 | 21840w3 | \([0, -1, 0, -290176, 229240960]\) | \(-659704930833045889/5156082432978750\) | \(-21119313645480960000\) | \([2]\) | \(491520\) | \(2.3910\) | |
| 21840.c1 | 21840w4 | \([0, -1, 0, -673856, 14572416]\) | \(8261629364934163009/4759323790524030\) | \(19494190245986426880\) | \([2]\) | \(491520\) | \(2.3910\) |
Rank
sage: E.rank()
The elliptic curves in class 21840w have rank \(0\).
Complex multiplication
The elliptic curves in class 21840w do not have complex multiplication.Modular form 21840.2.a.w
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.
