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SageMath
E = EllipticCurve("kq1")
E.isogeny_class()
Elliptic curves in class 463680.kq
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
463680.kq1 | 463680kq4 | \([0, 0, 0, -283370412, -1043471419184]\) | \(13167998447866683762601/5158996582031250000\) | \(985899600000000000000000000\) | \([2]\) | \(188743680\) | \(3.8783\) | |
463680.kq2 | 463680kq2 | \([0, 0, 0, -127573932, 543097614544]\) | \(1201550658189465626281/28577902500000000\) | \(5461322215587840000000000\) | \([2, 2]\) | \(94371840\) | \(3.5317\) | |
463680.kq3 | 463680kq1 | \([0, 0, 0, -126836652, 549813055696]\) | \(1180838681727016392361/692428800000\) | \(132325204348108800000\) | \([2]\) | \(47185920\) | \(3.1851\) | \(\Gamma_0(N)\)-optimal* |
463680.kq4 | 463680kq3 | \([0, 0, 0, 16426068, 1699878414544]\) | \(2564821295690373719/6533572090396050000\) | \(-1248585070385226173644800000\) | \([2]\) | \(188743680\) | \(3.8783\) |
Rank
sage: E.rank()
The elliptic curves in class 463680.kq have rank \(1\).
Complex multiplication
The elliptic curves in class 463680.kq do not have complex multiplication.Modular form 463680.2.a.kq
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.