Show commands:
SageMath
E = EllipticCurve("cw1")
E.isogeny_class()
Elliptic curves in class 221760.cw
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 221760.cw1 | 221760lt3 | \([0, 0, 0, -1182828, 495142832]\) | \(957681397954009/31185\) | \(5959546306560\) | \([2]\) | \(1572864\) | \(1.9526\) | |
| 221760.cw2 | 221760lt4 | \([0, 0, 0, -117228, -2325328]\) | \(932288503609/527295615\) | \(100767761258250240\) | \([2]\) | \(1572864\) | \(1.9526\) | |
| 221760.cw3 | 221760lt2 | \([0, 0, 0, -74028, 7714352]\) | \(234770924809/1334025\) | \(254936147558400\) | \([2, 2]\) | \(786432\) | \(1.6060\) | |
| 221760.cw4 | 221760lt1 | \([0, 0, 0, -2028, 255152]\) | \(-4826809/144375\) | \(-27590492160000\) | \([2]\) | \(393216\) | \(1.2595\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 221760.cw have rank \(1\).
Complex multiplication
The elliptic curves in class 221760.cw do not have complex multiplication.Modular form 221760.2.a.cw
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
