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SageMath
E = EllipticCurve("ii1")
E.isogeny_class()
Elliptic curves in class 284592.ii
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 284592.ii1 | 284592ii4 | \([0, 1, 0, -1340714888, 18893481708660]\) | \(312196988566716625/25367712678\) | \(21656367324211855639928832\) | \([2]\) | \(79626240\) | \(3.9066\) | |
| 284592.ii2 | 284592ii3 | \([0, 1, 0, -78075048, 337221564084]\) | \(-61653281712625/21875235228\) | \(-18674846069663738164396032\) | \([2]\) | \(39813120\) | \(3.5600\) | |
| 284592.ii3 | 284592ii2 | \([0, 1, 0, -34437608, -39024997068]\) | \(5290763640625/2291573592\) | \(1956311950105554983682048\) | \([2]\) | \(26542080\) | \(3.3573\) | |
| 284592.ii4 | 284592ii1 | \([0, 1, 0, 7302552, -4514232780]\) | \(50447927375/39517632\) | \(-33736126123709354999808\) | \([2]\) | \(13271040\) | \(3.0107\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 284592.ii have rank \(1\).
Complex multiplication
The elliptic curves in class 284592.ii do not have complex multiplication.Modular form 284592.2.a.ii
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
