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SageMath
E = EllipticCurve("en1")
E.isogeny_class()
Elliptic curves in class 259182.en
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 259182.en1 | 259182en4 | \([1, -1, 1, -90452000, 302099344363]\) | \(2347095854538583875/227772222179656\) | \(7942334287670381360843928\) | \([2]\) | \(66355200\) | \(3.5151\) | |
| 259182.en2 | 259182en3 | \([1, -1, 1, -88230440, 319008081835]\) | \(2178369958938055875/23936394944\) | \(834653358814757815872\) | \([2]\) | \(33177600\) | \(3.1686\) | |
| 259182.en3 | 259182en2 | \([1, -1, 1, -19516355, -33110655091]\) | \(17186994736845046875/41795067484786\) | \(1999147811807204215542\) | \([2]\) | \(22118400\) | \(2.9658\) | |
| 259182.en4 | 259182en1 | \([1, -1, 1, -1682165, -88868887]\) | \(11005489726162875/6296912450444\) | \(301194841975767623268\) | \([2]\) | \(11059200\) | \(2.6192\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 259182.en have rank \(1\).
Complex multiplication
The elliptic curves in class 259182.en do not have complex multiplication.Modular form 259182.2.a.en
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.
