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SageMath
E = EllipticCurve("bw1")
E.isogeny_class()
Elliptic curves in class 344850.bw
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
344850.bw1 | 344850bw3 | \([1, 1, 0, -3731166650, 87721476052500]\) | \(207530301091125281552569/805586668007040\) | \(22299155049394059210000000\) | \([2]\) | \(309657600\) | \(4.0771\) | |
344850.bw2 | 344850bw4 | \([1, 1, 0, -707134650, -5588609035500]\) | \(1412712966892699019449/330160465517040000\) | \(9139053194559889053750000000\) | \([2]\) | \(309657600\) | \(4.0771\) | |
344850.bw3 | 344850bw2 | \([1, 1, 0, -236686650, 1327447012500]\) | \(52974743974734147769/3152005008998400\) | \(87249517902284601600000000\) | \([2, 2]\) | \(154828800\) | \(3.7305\) | |
344850.bw4 | 344850bw1 | \([1, 1, 0, 11121350, 85681124500]\) | \(5495662324535111/117739817533440\) | \(-3259113576396226560000000\) | \([2]\) | \(77414400\) | \(3.3839\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 344850.bw have rank \(1\).
Complex multiplication
The elliptic curves in class 344850.bw do not have complex multiplication.Modular form 344850.2.a.bw
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.