Show commands:
SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 21.a
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 21.a1 | 21a5 | \([1, 0, 0, -784, -8515]\) | \(53297461115137/147\) | \(147\) | \([2]\) | \(4\) | \(0.074205\) | |
| 21.a2 | 21a2 | \([1, 0, 0, -49, -136]\) | \(13027640977/21609\) | \(21609\) | \([2, 2]\) | \(2\) | \(-0.27237\) | |
| 21.a3 | 21a3 | \([1, 0, 0, -39, 90]\) | \(6570725617/45927\) | \(45927\) | \([8]\) | \(2\) | \(-0.27237\) | |
| 21.a4 | 21a6 | \([1, 0, 0, -34, -217]\) | \(-4354703137/17294403\) | \(-17294403\) | \([2]\) | \(4\) | \(0.074205\) | |
| 21.a5 | 21a1 | \([1, 0, 0, -4, -1]\) | \(7189057/3969\) | \(3969\) | \([2, 4]\) | \(1\) | \(-0.61894\) | \(\Gamma_0(N)\)-optimal |
| 21.a6 | 21a4 | \([1, 0, 0, 1, 0]\) | \(103823/63\) | \(-63\) | \([4]\) | \(2\) | \(-0.96552\) |
Rank
sage: E.rank()
The elliptic curves in class 21.a have rank \(0\).
Complex multiplication
The elliptic curves in class 21.a do not have complex multiplication.Modular form 21.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 8 & 4 & 4 & 8 \\ 2 & 1 & 4 & 2 & 2 & 4 \\ 8 & 4 & 1 & 8 & 2 & 4 \\ 4 & 2 & 8 & 1 & 4 & 8 \\ 4 & 2 & 2 & 4 & 1 & 2 \\ 8 & 4 & 4 & 8 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
