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SageMath
E = EllipticCurve("bf1")
E.isogeny_class()
Elliptic curves in class 27225bf
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
27225.bb2 | 27225bf1 | \([0, 0, 1, -1397550, 1784829406]\) | \(-123633664/492075\) | \(-1201490487536354296875\) | \([]\) | \(912384\) | \(2.7302\) | \(\Gamma_0(N)\)-optimal |
27225.bb1 | 27225bf2 | \([0, 0, 1, -163114050, 801836784031]\) | \(-196566176333824/421875\) | \(-1030084437188232421875\) | \([]\) | \(2737152\) | \(3.2795\) |
Rank
sage: E.rank()
The elliptic curves in class 27225bf have rank \(1\).
Complex multiplication
The elliptic curves in class 27225bf do not have complex multiplication.Modular form 27225.2.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.
\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.