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SageMath
E = EllipticCurve("cd1")
E.isogeny_class()
Elliptic curves in class 42350.cd
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 42350.cd1 | 42350bu4 | \([1, -1, 1, -809755, 280653497]\) | \(2121328796049/120050\) | \(3323060907031250\) | \([2]\) | \(491520\) | \(2.0427\) | |
| 42350.cd2 | 42350bu3 | \([1, -1, 1, -265255, -49071503]\) | \(74565301329/5468750\) | \(151378503417968750\) | \([2]\) | \(491520\) | \(2.0427\) | |
| 42350.cd3 | 42350bu2 | \([1, -1, 1, -53505, 3865997]\) | \(611960049/122500\) | \(3390878476562500\) | \([2, 2]\) | \(245760\) | \(1.6961\) | |
| 42350.cd4 | 42350bu1 | \([1, -1, 1, 6995, 356997]\) | \(1367631/2800\) | \(-77505793750000\) | \([2]\) | \(122880\) | \(1.3495\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 42350.cd have rank \(1\).
Complex multiplication
The elliptic curves in class 42350.cd do not have complex multiplication.Modular form 42350.2.a.cd
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.
