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SageMath
E = EllipticCurve("bd1")
E.isogeny_class()
Elliptic curves in class 68400.bd
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
68400.bd1 | 68400cn2 | \([0, 0, 0, -14923875, 19408131250]\) | \(252122146858292/34296447249\) | \(50004220089042000000000\) | \([2]\) | \(3686400\) | \(3.0826\) | |
68400.bd2 | 68400cn1 | \([0, 0, 0, 1478625, 1611418750]\) | \(980844844912/3645153819\) | \(-1328658567025500000000\) | \([2]\) | \(1843200\) | \(2.7361\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 68400.bd have rank \(1\).
Complex multiplication
The elliptic curves in class 68400.bd do not have complex multiplication.Modular form 68400.2.a.bd
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.