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SageMath
E = EllipticCurve("cf1")
E.isogeny_class()
Elliptic curves in class 108900cf
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
108900.x2 | 108900cf1 | \([0, 0, 0, 72600, -4159375]\) | \(131072/99\) | \(-31963832232750000\) | \([2]\) | \(737280\) | \(1.8551\) | \(\Gamma_0(N)\)-optimal |
108900.x1 | 108900cf2 | \([0, 0, 0, -335775, -35604250]\) | \(810448/363\) | \(1875211490988000000\) | \([2]\) | \(1474560\) | \(2.2017\) |
Rank
sage: E.rank()
The elliptic curves in class 108900cf have rank \(2\).
Complex multiplication
The elliptic curves in class 108900cf do not have complex multiplication.Modular form 108900.2.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.