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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 450840.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
450840.a1 | 450840a4 | \([0, -1, 0, -162302496, -795804498180]\) | \(19129597231400697604/26325\) | \(650671620019200\) | \([2]\) | \(31457280\) | \(3.0087\) | |
450840.a2 | 450840a2 | \([0, -1, 0, -10143996, -12431676780]\) | \(18681746265374416/693005625\) | \(4282232599251360000\) | \([2, 2]\) | \(15728640\) | \(2.6621\) | |
450840.a3 | 450840a3 | \([0, -1, 0, -9675816, -13631528484]\) | \(-4053153720264484/903687890625\) | \(-22336336705971600000000\) | \([2]\) | \(31457280\) | \(3.0087\) | |
450840.a4 | 450840a1 | \([0, -1, 0, -663351, -175098924]\) | \(83587439220736/13990184325\) | \(5403024631478494800\) | \([2]\) | \(7864320\) | \(2.3156\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 450840.a have rank \(0\).
Complex multiplication
The elliptic curves in class 450840.a do not have complex multiplication.Modular form 450840.2.a.a
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.