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SageMath
E = EllipticCurve("bk1")
E.isogeny_class()
Elliptic curves in class 480240.bk
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
480240.bk1 | 480240bk4 | \([0, 0, 0, -192096003, -1024768119998]\) | \(262537424941059264096001/250125\) | \(746869248000\) | \([2]\) | \(25165824\) | \(2.9545\) | |
480240.bk2 | 480240bk2 | \([0, 0, 0, -12006003, -16011993998]\) | \(64096096056024006001/62562515625\) | \(186810670656000000\) | \([2, 2]\) | \(12582912\) | \(2.6079\) | |
480240.bk3 | 480240bk3 | \([0, 0, 0, -11916003, -16263867998]\) | \(-62665433378363916001/2004003001000125\) | \(-5983920896938357248000\) | \([2]\) | \(25165824\) | \(2.9545\) | |
480240.bk4 | 480240bk1 | \([0, 0, 0, -756003, -246243998]\) | \(16003198512756001/488525390625\) | \(1458729000000000000\) | \([2]\) | \(6291456\) | \(2.2613\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 480240.bk have rank \(1\).
Complex multiplication
The elliptic curves in class 480240.bk do not have complex multiplication.Modular form 480240.2.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 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.