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SageMath
E = EllipticCurve("bd1")
E.isogeny_class()
Elliptic curves in class 142912.bd
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
142912.bd1 | 142912o2 | \([0, 0, 0, -112300, 14227152]\) | \(597479568890625/12199182688\) | \(3197942546563072\) | \([2]\) | \(491520\) | \(1.7658\) | |
142912.bd2 | 142912o1 | \([0, 0, 0, 340, 665296]\) | \(16581375/729422848\) | \(-191213823066112\) | \([2]\) | \(245760\) | \(1.4193\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 142912.bd have rank \(1\).
Complex multiplication
The elliptic curves in class 142912.bd do not have complex multiplication.Modular form 142912.2.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.