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SageMath
E = EllipticCurve("er1")
E.isogeny_class()
Elliptic curves in class 53550.er
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
53550.er1 | 53550eo2 | \([1, -1, 1, -258305, 299023697]\) | \(-6693187811305/131714173248\) | \(-37507668866325000000\) | \([3]\) | \(1555200\) | \(2.4369\) | |
53550.er2 | 53550eo1 | \([1, -1, 1, 28570, -10801303]\) | \(9056932295/181997172\) | \(-51826538432812500\) | \([]\) | \(518400\) | \(1.8876\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 53550.er have rank \(1\).
Complex multiplication
The elliptic curves in class 53550.er do not have complex multiplication.Modular form 53550.2.a.er
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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.