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SageMath
E = EllipticCurve("ck1")
E.isogeny_class()
Elliptic curves in class 21294.ck
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
21294.ck1 | 21294cg4 | \([1, -1, 1, -634289, -194274349]\) | \(8020417344913/187278\) | \(658983294072558\) | \([2]\) | \(258048\) | \(1.9558\) | |
21294.ck2 | 21294cg2 | \([1, -1, 1, -41099, -2792617]\) | \(2181825073/298116\) | \(1048993815054276\) | \([2, 2]\) | \(129024\) | \(1.6092\) | |
21294.ck3 | 21294cg1 | \([1, -1, 1, -10679, 383231]\) | \(38272753/4368\) | \(15369872748048\) | \([4]\) | \(64512\) | \(1.2626\) | \(\Gamma_0(N)\)-optimal |
21294.ck4 | 21294cg3 | \([1, -1, 1, 65371, -14930197]\) | \(8780064047/32388174\) | \(-113965685192682414\) | \([2]\) | \(258048\) | \(1.9558\) |
Rank
sage: E.rank()
The elliptic curves in class 21294.ck have rank \(1\).
Complex multiplication
The elliptic curves in class 21294.ck do not have complex multiplication.Modular form 21294.2.a.ck
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.