Show commands:
SageMath
E = EllipticCurve("cw1")
E.isogeny_class()
Elliptic curves in class 112710.cw
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 112710.cw1 | 112710ct4 | \([1, 0, 0, -109042596, -438279542064]\) | \(5940441603429810927841/3044264109120\) | \(73481134988107529280\) | \([2]\) | \(21233664\) | \(3.1442\) | |
| 112710.cw2 | 112710ct2 | \([1, 0, 0, -6852196, -6770359024]\) | \(1474074790091785441/32813650022400\) | \(792041741557531545600\) | \([2, 2]\) | \(10616832\) | \(2.7976\) | |
| 112710.cw3 | 112710ct1 | \([1, 0, 0, -933476, 191239440]\) | \(3726830856733921/1501644718080\) | \(36246052996141547520\) | \([2]\) | \(5308416\) | \(2.4510\) | \(\Gamma_0(N)\)-optimal |
| 112710.cw4 | 112710ct3 | \([1, 0, 0, 638684, -20779802800]\) | \(1193680917131039/7728836230440000\) | \(-186555317801945400360000\) | \([2]\) | \(21233664\) | \(3.1442\) |
Rank
sage: E.rank()
The elliptic curves in class 112710.cw have rank \(0\).
Complex multiplication
The elliptic curves in class 112710.cw do not have complex multiplication.Modular form 112710.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
