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SageMath
E = EllipticCurve("bf1")
E.isogeny_class()
Elliptic curves in class 127400bf
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
127400.bd1 | 127400bf1 | \([0, 0, 0, -3365075, 117734750]\) | \(2238719766084/1292374265\) | \(2432744638447760000000\) | \([2]\) | \(3981312\) | \(2.7931\) | \(\Gamma_0(N)\)-optimal |
127400.bd2 | 127400bf2 | \([0, 0, 0, 13441925, 941277750]\) | \(71346044015118/41389887175\) | \(-155823322760050400000000\) | \([2]\) | \(7962624\) | \(3.1397\) |
Rank
sage: E.rank()
The elliptic curves in class 127400bf have rank \(1\).
Complex multiplication
The elliptic curves in class 127400bf do not have complex multiplication.Modular form 127400.2.a.bf
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.