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SageMath
E = EllipticCurve("bj1")
E.isogeny_class()
Elliptic curves in class 177744.bj
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
177744.bj1 | 177744cb4 | \([0, -1, 0, -1513778112, -22668969536640]\) | \(632678989847546725777/80515134\) | \(48820754184782340096\) | \([2]\) | \(48660480\) | \(3.6379\) | |
177744.bj2 | 177744cb3 | \([0, -1, 0, -108246272, -245436938112]\) | \(231331938231569617/90942310746882\) | \(55143324955152100753809408\) | \([2]\) | \(48660480\) | \(3.6379\) | |
177744.bj3 | 177744cb2 | \([0, -1, 0, -94619232, -354115307520]\) | \(154502321244119857/55101928644\) | \(33411330016989612097536\) | \([2, 2]\) | \(24330240\) | \(3.2913\) | |
177744.bj4 | 177744cb1 | \([0, -1, 0, -5070112, -7166196992]\) | \(-23771111713777/22848457968\) | \(-13854276761894967508992\) | \([2]\) | \(12165120\) | \(2.9447\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 177744.bj have rank \(1\).
Complex multiplication
The elliptic curves in class 177744.bj do not have complex multiplication.Modular form 177744.2.a.bj
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.