Show commands:
SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 1027.a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 1027.a1 | 1027a1 | \([0, 1, 1, -213, 1128]\) | \(-1073741824000/1027\) | \(-1027\) | \([3]\) | \(128\) | \(-0.12713\) | \(\Gamma_0(N)\)-optimal |
| 1027.a2 | 1027a2 | \([0, 1, 1, -163, 1721]\) | \(-481890304000/1083206683\) | \(-1083206683\) | \([3]\) | \(384\) | \(0.42218\) | |
| 1027.a3 | 1027a3 | \([0, 1, 1, 1417, -38490]\) | \(314432000000000/837755450467\) | \(-837755450467\) | \([]\) | \(1152\) | \(0.97149\) |
Rank
sage: E.rank()
The elliptic curves in class 1027.a have rank \(1\).
Complex multiplication
The elliptic curves in class 1027.a do not have complex multiplication.Modular form 1027.2.a.a
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.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
