Show commands:
SageMath
E = EllipticCurve("z1")
E.isogeny_class()
Elliptic curves in class 44880z
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 44880.i3 | 44880z1 | \([0, -1, 0, -219656, 25041456]\) | \(286150792766867209/99845947265625\) | \(408969000000000000\) | \([2]\) | \(559104\) | \(2.0809\) | \(\Gamma_0(N)\)-optimal |
| 44880.i2 | 44880z2 | \([0, -1, 0, -1469656, -666958544]\) | \(85705982088578117209/2613369421265625\) | \(10704361149504000000\) | \([2, 2]\) | \(1118208\) | \(2.4274\) | |
| 44880.i4 | 44880z3 | \([0, -1, 0, 400344, -2249726544]\) | \(1732457747755512791/534745023634713375\) | \(-2190315616807785984000\) | \([2]\) | \(2236416\) | \(2.7740\) | |
| 44880.i1 | 44880z4 | \([0, -1, 0, -23339656, -43392190544]\) | \(343278919869647291747209/334291413963375\) | \(1369257631593984000\) | \([2]\) | \(2236416\) | \(2.7740\) |
Rank
sage: E.rank()
The elliptic curves in class 44880z have rank \(0\).
Complex multiplication
The elliptic curves in class 44880z do not have complex multiplication.Modular form 44880.2.a.z
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 Cremona 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 Cremona labels.
