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SageMath
E = EllipticCurve("kj1")
E.isogeny_class()
Elliptic curves in class 463680.kj
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
463680.kj1 | 463680kj4 | \([0, 0, 0, -6677292, -6641232176]\) | \(344577854816148242/2716875\) | \(259601448960000\) | \([2]\) | \(7864320\) | \(2.3568\) | |
463680.kj2 | 463680kj2 | \([0, 0, 0, -417612, -103622384]\) | \(168591300897604/472410225\) | \(22569749972582400\) | \([2, 2]\) | \(3932160\) | \(2.0102\) | |
463680.kj3 | 463680kj3 | \([0, 0, 0, -252012, -186621104]\) | \(-18524646126002/146738831715\) | \(-14021113717749841920\) | \([2]\) | \(7864320\) | \(2.3568\) | |
463680.kj4 | 463680kj1 | \([0, 0, 0, -36732, -175376]\) | \(458891455696/264449745\) | \(3158570829496320\) | \([2]\) | \(1966080\) | \(1.6636\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 463680.kj have rank \(0\).
Complex multiplication
The elliptic curves in class 463680.kj do not have complex multiplication.Modular form 463680.2.a.kj
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.