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SageMath
E = EllipticCurve("dx1")
E.isogeny_class()
Elliptic curves in class 493680.dx
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
493680.dx1 | 493680dx4 | \([0, 1, 0, -12663416, -17348995116]\) | \(30949975477232209/478125000\) | \(3469425062400000000\) | \([2]\) | \(23592960\) | \(2.6926\) | |
493680.dx2 | 493680dx2 | \([0, 1, 0, -815096, -254239020]\) | \(8253429989329/936360000\) | \(6794522042204160000\) | \([2, 2]\) | \(11796480\) | \(2.3461\) | |
493680.dx3 | 493680dx1 | \([0, 1, 0, -195576, 29005524]\) | \(114013572049/15667200\) | \(113686120444723200\) | \([2]\) | \(5898240\) | \(1.9995\) | \(\Gamma_0(N)\)-optimal* |
493680.dx4 | 493680dx3 | \([0, 1, 0, 1120904, -1277221420]\) | \(21464092074671/109596256200\) | \(-795264832429785907200\) | \([2]\) | \(23592960\) | \(2.6926\) |
Rank
sage: E.rank()
The elliptic curves in class 493680.dx have rank \(2\).
Complex multiplication
The elliptic curves in class 493680.dx do not have complex multiplication.Modular form 493680.2.a.dx
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.