Show commands:
SageMath
E = EllipticCurve("i1")
E.isogeny_class()
Elliptic curves in class 263568.i
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 263568.i1 | 263568i3 | \([0, -1, 0, -404840544, 3135398932224]\) | \(74220219816682217473/16416\) | \(1623008594755584\) | \([2]\) | \(26542080\) | \(3.2112\) | |
| 263568.i2 | 263568i2 | \([0, -1, 0, -25302624, 48996566784]\) | \(18120364883707393/269485056\) | \(26643309091507666944\) | \([2, 2]\) | \(13271040\) | \(2.8646\) | |
| 263568.i3 | 263568i4 | \([0, -1, 0, -24562784, 51995582208]\) | \(-16576888679672833/2216253521952\) | \(-219115406571968162562048\) | \([2]\) | \(26542080\) | \(3.2112\) | |
| 263568.i4 | 263568i1 | \([0, -1, 0, -1627744, 718751488]\) | \(4824238966273/537919488\) | \(53182745632950976512\) | \([2]\) | \(6635520\) | \(2.5180\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 263568.i have rank \(1\).
Complex multiplication
The elliptic curves in class 263568.i do not have complex multiplication.Modular form 263568.2.a.i
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.
