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SageMath
E = EllipticCurve("fk1")
E.isogeny_class()
Elliptic curves in class 224400fk
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
224400.cn1 | 224400fk1 | \([0, -1, 0, -3608, 41712]\) | \(81182737/35904\) | \(2297856000000\) | \([2]\) | \(294912\) | \(1.0676\) | \(\Gamma_0(N)\)-optimal |
224400.cn2 | 224400fk2 | \([0, -1, 0, 12392, 297712]\) | \(3288008303/2517768\) | \(-161137152000000\) | \([2]\) | \(589824\) | \(1.4142\) |
Rank
sage: E.rank()
The elliptic curves in class 224400fk have rank \(1\).
Complex multiplication
The elliptic curves in class 224400fk do not have complex multiplication.Modular form 224400.2.a.fk
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.