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SageMath
E = EllipticCurve("a1")
E.isogeny_class()
Elliptic curves in class 259920.a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
259920.a1 | 259920a3 | \([0, 0, 0, -106698243, 424212949042]\) | \(3825131988299044/961875\) | \(33780628682375040000\) | \([2]\) | \(29491200\) | \(3.1224\) | |
259920.a2 | 259920a2 | \([0, 0, 0, -6694023, 6575325478]\) | \(3778298043856/59213025\) | \(519883875421751865600\) | \([2, 2]\) | \(14745600\) | \(2.7758\) | |
259920.a3 | 259920a1 | \([0, 0, 0, -829578, -132426713]\) | \(115060504576/52780005\) | \(28962666516551299920\) | \([2]\) | \(7372800\) | \(2.4292\) | \(\Gamma_0(N)\)-optimal |
259920.a4 | 259920a4 | \([0, 0, 0, -520923, 18233842138]\) | \(-445138564/4089438495\) | \(-143619288700720589061120\) | \([2]\) | \(29491200\) | \(3.1224\) |
Rank
sage: E.rank()
The elliptic curves in class 259920.a have rank \(1\).
Complex multiplication
The elliptic curves in class 259920.a do not have complex multiplication.Modular form 259920.2.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.