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SageMath
E = EllipticCurve("ex1")
E.isogeny_class()
Elliptic curves in class 223440ex
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 223440.bg4 | 223440ex1 | \([0, -1, 0, -1489616, 731868096]\) | \(-758575480593601/40535043840\) | \(-19533444598710927360\) | \([2]\) | \(6635520\) | \(2.4604\) | \(\Gamma_0(N)\)-optimal |
| 223440.bg3 | 223440ex2 | \([0, -1, 0, -24131536, 45635323840]\) | \(3225005357698077121/8526675600\) | \(4108922296993382400\) | \([2, 2]\) | \(13271040\) | \(2.8070\) | |
| 223440.bg1 | 223440ex3 | \([0, -1, 0, -386104336, 2920278512320]\) | \(13209596798923694545921/92340\) | \(44497750671360\) | \([2]\) | \(26542080\) | \(3.1536\) | |
| 223440.bg2 | 223440ex4 | \([0, -1, 0, -24429456, 44451032256]\) | \(3345930611358906241/165622259047500\) | \(79811760761566525440000\) | \([2]\) | \(26542080\) | \(3.1536\) |
Rank
sage: E.rank()
The elliptic curves in class 223440ex have rank \(0\).
Complex multiplication
The elliptic curves in class 223440ex do not have complex multiplication.Modular form 223440.2.a.ex
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.
