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SageMath
E = EllipticCurve("bm1")
E.isogeny_class()
Elliptic curves in class 207025bm
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
207025.bm2 | 207025bm1 | \([0, -1, 1, -690083, -204734307]\) | \(163840/13\) | \(2883711045520703125\) | \([]\) | \(2903040\) | \(2.2857\) | \(\Gamma_0(N)\)-optimal |
207025.bm1 | 207025bm2 | \([0, -1, 1, -11041333, 14085166318]\) | \(671088640/2197\) | \(487347166692998828125\) | \([]\) | \(8709120\) | \(2.8350\) |
Rank
sage: E.rank()
The elliptic curves in class 207025bm have rank \(1\).
Complex multiplication
The elliptic curves in class 207025bm do not have complex multiplication.Modular form 207025.2.a.bm
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.