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SageMath
E = EllipticCurve("bk1")
E.isogeny_class()
Elliptic curves in class 52800bk
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
52800.h3 | 52800bk1 | \([0, -1, 0, -27633, 1777137]\) | \(9115564624/825\) | \(211200000000\) | \([2]\) | \(147456\) | \(1.2125\) | \(\Gamma_0(N)\)-optimal |
52800.h2 | 52800bk2 | \([0, -1, 0, -29633, 1507137]\) | \(2810381476/680625\) | \(696960000000000\) | \([2, 2]\) | \(294912\) | \(1.5591\) | |
52800.h4 | 52800bk3 | \([0, -1, 0, 70367, 9407137]\) | \(18814587262/29648025\) | \(-60719155200000000\) | \([2]\) | \(589824\) | \(1.9056\) | |
52800.h1 | 52800bk4 | \([0, -1, 0, -161633, -23704863]\) | \(228027144098/12890625\) | \(26400000000000000\) | \([2]\) | \(589824\) | \(1.9056\) |
Rank
sage: E.rank()
The elliptic curves in class 52800bk have rank \(0\).
Complex multiplication
The elliptic curves in class 52800bk do not have complex multiplication.Modular form 52800.2.a.bk
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 Cremona 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 Cremona labels.