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SageMath
E = EllipticCurve("cf1")
E.isogeny_class()
Elliptic curves in class 54208cf
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
54208.cx2 | 54208cf1 | \([0, -1, 0, 26943, 6450017]\) | \(4657463/41503\) | \(-19274162813796352\) | \([2]\) | \(368640\) | \(1.8061\) | \(\Gamma_0(N)\)-optimal |
54208.cx1 | 54208cf2 | \([0, -1, 0, -398977, 89674785]\) | \(15124197817/1294139\) | \(601003440466558976\) | \([2]\) | \(737280\) | \(2.1526\) |
Rank
sage: E.rank()
The elliptic curves in class 54208cf have rank \(1\).
Complex multiplication
The elliptic curves in class 54208cf do not have complex multiplication.Modular form 54208.2.a.cf
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.