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SageMath
E = EllipticCurve("bc1")
E.isogeny_class()
Elliptic curves in class 223146.bc
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
223146.bc1 | 223146de2 | \([1, -1, 0, -27371997, 55126669639]\) | \(26444015547214434625/46191222\) | \(3961641935189862\) | \([2]\) | \(8257536\) | \(2.6808\) | |
223146.bc2 | 223146de1 | \([1, -1, 0, -1710207, 862248505]\) | \(-6449916994998625/8532911772\) | \(-731834743519676412\) | \([2]\) | \(4128768\) | \(2.3343\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 223146.bc have rank \(1\).
Complex multiplication
The elliptic curves in class 223146.bc do not have complex multiplication.Modular form 223146.2.a.bc
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.