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SageMath
E = EllipticCurve("cd1")
E.isogeny_class()
Elliptic curves in class 169050.cd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 169050.cd1 | 169050fr4 | \([1, 0, 1, -590891026, 5410558439948]\) | \(12411881707829361287041/303132494474220600\) | \(557238044412462177646875000\) | \([2]\) | \(143327232\) | \(3.9162\) | |
| 169050.cd2 | 169050fr2 | \([1, 0, 1, -72716026, -235785060052]\) | \(23131609187144855041/322060536000000\) | \(592032812497875000000000\) | \([2]\) | \(47775744\) | \(3.3669\) | |
| 169050.cd3 | 169050fr1 | \([1, 0, 1, -588026, -9880164052]\) | \(-12232183057921/22933241856000\) | \(-42157390173696000000000\) | \([2]\) | \(23887872\) | \(3.0203\) | \(\Gamma_0(N)\)-optimal |
| 169050.cd4 | 169050fr3 | \([1, 0, 1, 5291974, 266691515948]\) | \(8915971454369279/16719623332762560\) | \(-30735108835565350335000000\) | \([2]\) | \(71663616\) | \(3.5696\) |
Rank
sage: E.rank()
The elliptic curves in class 169050.cd have rank \(0\).
Complex multiplication
The elliptic curves in class 169050.cd do not have complex multiplication.Modular form 169050.2.a.cd
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
