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SageMath
E = EllipticCurve("co1")
E.isogeny_class()
Elliptic curves in class 72450.co
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
72450.co1 | 72450ew2 | \([1, -1, 1, -249260, 45256367]\) | \(18794999107081133/1193761408896\) | \(108781508385648000\) | \([2]\) | \(946176\) | \(2.0198\) | |
72450.co2 | 72450ew1 | \([1, -1, 1, -47660, -3127633]\) | \(131383171726253/29303586816\) | \(2670289348608000\) | \([2]\) | \(473088\) | \(1.6732\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 72450.co have rank \(1\).
Complex multiplication
The elliptic curves in class 72450.co do not have complex multiplication.Modular form 72450.2.a.co
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 LMFDB 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 LMFDB labels.