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SageMath
E = EllipticCurve("cw1")
E.isogeny_class()
Elliptic curves in class 25200.cw
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 25200.cw1 | 25200dy4 | \([0, 0, 0, -963675, -364101750]\) | \(2121328796049/120050\) | \(5601052800000000\) | \([2]\) | \(294912\) | \(2.0862\) | |
| 25200.cw2 | 25200dy3 | \([0, 0, 0, -315675, 63794250]\) | \(74565301329/5468750\) | \(255150000000000000\) | \([2]\) | \(294912\) | \(2.0862\) | |
| 25200.cw3 | 25200dy2 | \([0, 0, 0, -63675, -5001750]\) | \(611960049/122500\) | \(5715360000000000\) | \([2, 2]\) | \(147456\) | \(1.7396\) | |
| 25200.cw4 | 25200dy1 | \([0, 0, 0, 8325, -465750]\) | \(1367631/2800\) | \(-130636800000000\) | \([2]\) | \(73728\) | \(1.3930\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 25200.cw have rank \(1\).
Complex multiplication
The elliptic curves in class 25200.cw do not have complex multiplication.Modular form 25200.2.a.cw
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.
