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SageMath
E = EllipticCurve("cw1")
E.isogeny_class()
Elliptic curves in class 242550cw
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
242550.cw4 | 242550cw1 | \([1, -1, 0, 38358, -1771484]\) | \(4657463/3696\) | \(-4952993487750000\) | \([2]\) | \(1572864\) | \(1.6995\) | \(\Gamma_0(N)\)-optimal |
242550.cw3 | 242550cw2 | \([1, -1, 0, -182142, -15221984]\) | \(498677257/213444\) | \(286035373917562500\) | \([2, 2]\) | \(3145728\) | \(2.0461\) | |
242550.cw2 | 242550cw3 | \([1, -1, 0, -1394892, 623897266]\) | \(223980311017/4278582\) | \(5733709086256593750\) | \([2]\) | \(6291456\) | \(2.3926\) | |
242550.cw1 | 242550cw4 | \([1, -1, 0, -2497392, -1517819234]\) | \(1285429208617/614922\) | \(824054291524406250\) | \([2]\) | \(6291456\) | \(2.3926\) |
Rank
sage: E.rank()
The elliptic curves in class 242550cw have rank \(1\).
Complex multiplication
The elliptic curves in class 242550cw do not have complex multiplication.Modular form 242550.2.a.cw
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 Cremona 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 Cremona labels.