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SageMath
E = EllipticCurve("cf1")
E.isogeny_class()
Elliptic curves in class 54450cf
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
54450.cv2 | 54450cf1 | \([1, -1, 0, 271683, -220093659]\) | \(109902239/1100000\) | \(-22197105717187500000\) | \([]\) | \(1728000\) | \(2.3933\) | \(\Gamma_0(N)\)-optimal |
54450.cv1 | 54450cf2 | \([1, -1, 0, -161717067, -791516079909]\) | \(-23178622194826561/1610510\) | \(-32498782480534218750\) | \([]\) | \(8640000\) | \(3.1980\) |
Rank
sage: E.rank()
The elliptic curves in class 54450cf have rank \(1\).
Complex multiplication
The elliptic curves in class 54450cf do not have complex multiplication.Modular form 54450.2.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.
\(\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.