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SageMath
E = EllipticCurve("ej1")
E.isogeny_class()
Elliptic curves in class 33600ej
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
33600.bm2 | 33600ej1 | \([0, -1, 0, 2592, -88938]\) | \(1925134784/4465125\) | \(-4465125000000\) | \([2]\) | \(55296\) | \(1.1113\) | \(\Gamma_0(N)\)-optimal |
33600.bm1 | 33600ej2 | \([0, -1, 0, -21033, -963063]\) | \(16079333824/2953125\) | \(189000000000000\) | \([2]\) | \(110592\) | \(1.4579\) |
Rank
sage: E.rank()
The elliptic curves in class 33600ej have rank \(0\).
Complex multiplication
The elliptic curves in class 33600ej do not have complex multiplication.Modular form 33600.2.a.ej
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 Cremona numbering.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.