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SageMath
E = EllipticCurve("x1")
E.isogeny_class()
Elliptic curves in class 34650.x
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 34650.x1 | 34650w4 | \([1, -1, 0, -4704567, 3928779341]\) | \(1010962818911303721/57392720\) | \(653738951250000\) | \([2]\) | \(786432\) | \(2.3094\) | |
| 34650.x2 | 34650w3 | \([1, -1, 0, -492567, -31256659]\) | \(1160306142246441/634128110000\) | \(7223115502968750000\) | \([2]\) | \(786432\) | \(2.3094\) | |
| 34650.x3 | 34650w2 | \([1, -1, 0, -294567, 61209341]\) | \(248158561089321/1859334400\) | \(21178980900000000\) | \([2, 2]\) | \(393216\) | \(1.9628\) | |
| 34650.x4 | 34650w1 | \([1, -1, 0, -6567, 2169341]\) | \(-2749884201/176619520\) | \(-2011806720000000\) | \([2]\) | \(196608\) | \(1.6163\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 34650.x have rank \(1\).
Complex multiplication
The elliptic curves in class 34650.x do not have complex multiplication.Modular form 34650.2.a.x
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.
