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SageMath
E = EllipticCurve("dm1")
E.isogeny_class()
Elliptic curves in class 388080.dm
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 388080.dm1 | 388080dm3 | \([0, 0, 0, -171185763, 862083966562]\) | \(1579250141304807889/41926500\) | \(14728696922548224000\) | \([2]\) | \(47775744\) | \(3.1921\) | |
| 388080.dm2 | 388080dm4 | \([0, 0, 0, -170974083, 864322313218]\) | \(-1573398910560073969/8138108343750\) | \(-2858901442237120896000000\) | \([2]\) | \(95551488\) | \(3.5387\) | |
| 388080.dm3 | 388080dm1 | \([0, 0, 0, -2265123, 1003002658]\) | \(3658671062929/880165440\) | \(309200386568430551040\) | \([2]\) | \(15925248\) | \(2.6428\) | \(\Gamma_0(N)\)-optimal |
| 388080.dm4 | 388080dm2 | \([0, 0, 0, 5355357, 6305332642]\) | \(48351870250991/76871856600\) | \(-27004931910247418265600\) | \([2]\) | \(31850496\) | \(2.9894\) |
Rank
sage: E.rank()
The elliptic curves in class 388080.dm have rank \(0\).
Complex multiplication
The elliptic curves in class 388080.dm do not have complex multiplication.Modular form 388080.2.a.dm
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.
