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SageMath
E = EllipticCurve("cz1")
E.isogeny_class()
Elliptic curves in class 206910.cz
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 206910.cz1 | 206910bf3 | \([1, -1, 1, -536310743, 4780628964411]\) | \(13209596798923694545921/92340\) | \(119254152257460\) | \([2]\) | \(44236800\) | \(3.2357\) | |
| 206910.cz2 | 206910bf4 | \([1, -1, 1, -33933263, 72764683467]\) | \(3345930611358906241/165622259047500\) | \(213895842513266699527500\) | \([2]\) | \(44236800\) | \(3.2357\) | |
| 206910.cz3 | 206910bf2 | \([1, -1, 1, -33519443, 74703512931]\) | \(3225005357698077121/8526675600\) | \(11011928419453856400\) | \([2, 2]\) | \(22118400\) | \(2.8892\) | |
| 206910.cz4 | 206910bf1 | \([1, -1, 1, -2069123, 1197825027]\) | \(-758575480593601/40535043840\) | \(-52349710741370760960\) | \([2]\) | \(11059200\) | \(2.5426\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 206910.cz have rank \(0\).
Complex multiplication
The elliptic curves in class 206910.cz do not have complex multiplication.Modular form 206910.2.a.cz
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
