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SageMath
E = EllipticCurve("cf1")
E.isogeny_class()
Elliptic curves in class 35904.cf
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
35904.cf1 | 35904bg4 | \([0, 1, 0, -123969, -15612129]\) | \(803760366578833/65593817586\) | \(17195025717264384\) | \([2]\) | \(294912\) | \(1.8580\) | |
35904.cf2 | 35904bg2 | \([0, 1, 0, -26049, 1328031]\) | \(7457162887153/1370924676\) | \(359379678265344\) | \([2, 2]\) | \(147456\) | \(1.5114\) | |
35904.cf3 | 35904bg1 | \([0, 1, 0, -24769, 1492127]\) | \(6411014266033/296208\) | \(77649149952\) | \([2]\) | \(73728\) | \(1.1648\) | \(\Gamma_0(N)\)-optimal |
35904.cf4 | 35904bg3 | \([0, 1, 0, 51391, 7786527]\) | \(57258048889007/132611470002\) | \(-34763301192204288\) | \([4]\) | \(294912\) | \(1.8580\) |
Rank
sage: E.rank()
The elliptic curves in class 35904.cf have rank \(1\).
Complex multiplication
The elliptic curves in class 35904.cf do not have complex multiplication.Modular form 35904.2.a.cf
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.