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SageMath
E = EllipticCurve("bo1")
E.isogeny_class()
Elliptic curves in class 17640bo
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
17640.k2 | 17640bo1 | \([0, 0, 0, -241184223, 1441688560578]\) | \(7630566466251024/78125\) | \(15885600931620000000\) | \([2]\) | \(1806336\) | \(3.2586\) | \(\Gamma_0(N)\)-optimal |
17640.k1 | 17640bo2 | \([0, 0, 0, -241369443, 1439363345742]\) | \(1912039973861076/6103515625\) | \(4964250291131250000000000\) | \([2]\) | \(3612672\) | \(3.6052\) |
Rank
sage: E.rank()
The elliptic curves in class 17640bo have rank \(1\).
Complex multiplication
The elliptic curves in class 17640bo do not have complex multiplication.Modular form 17640.2.a.bo
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.