Show commands:
SageMath
E = EllipticCurve("ef1")
E.isogeny_class()
Elliptic curves in class 346560.ef
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 346560.ef1 | 346560ef3 | \([0, -1, 0, -14348065, 20893240225]\) | \(26487576322129/44531250\) | \(549194796441600000000\) | \([2]\) | \(17694720\) | \(2.8749\) | |
| 346560.ef2 | 346560ef2 | \([0, -1, 0, -1178785, 104214817]\) | \(14688124849/8122500\) | \(100173130870947840000\) | \([2, 2]\) | \(8847360\) | \(2.5283\) | |
| 346560.ef3 | 346560ef1 | \([0, -1, 0, -716705, -231902175]\) | \(3301293169/22800\) | \(281187735778099200\) | \([2]\) | \(4423680\) | \(2.1817\) | \(\Gamma_0(N)\)-optimal |
| 346560.ef4 | 346560ef4 | \([0, -1, 0, 4597215, 819283617]\) | \(871257511151/527800050\) | \(-6509250043994190643200\) | \([2]\) | \(17694720\) | \(2.8749\) |
Rank
sage: E.rank()
The elliptic curves in class 346560.ef have rank \(1\).
Complex multiplication
The elliptic curves in class 346560.ef do not have complex multiplication.Modular form 346560.2.a.ef
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.
