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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 4005.d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 4005.d1 | 4005c3 | \([1, -1, 0, -21375, 1208196]\) | \(1481582988342001/2919645\) | \(2128421205\) | \([2]\) | \(7168\) | \(1.0424\) | |
| 4005.d2 | 4005c2 | \([1, -1, 0, -1350, 18711]\) | \(373403541601/16040025\) | \(11693178225\) | \([2, 2]\) | \(3584\) | \(0.69580\) | |
| 4005.d3 | 4005c1 | \([1, -1, 0, -225, -864]\) | \(1732323601/500625\) | \(364955625\) | \([2]\) | \(1792\) | \(0.34922\) | \(\Gamma_0(N)\)-optimal |
| 4005.d4 | 4005c4 | \([1, -1, 0, 675, 68526]\) | \(46617130799/2823400845\) | \(-2058259216005\) | \([2]\) | \(7168\) | \(1.0424\) |
Rank
sage: E.rank()
The elliptic curves in class 4005.d have rank \(0\).
Complex multiplication
The elliptic curves in class 4005.d do not have complex multiplication.Modular form 4005.2.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.
