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SageMath
E = EllipticCurve("cu1")
E.isogeny_class()
Elliptic curves in class 221760.cu
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
221760.cu1 | 221760lr4 | \([0, 0, 0, -15001068, -22361378128]\) | \(1953542217204454969/170843779260\) | \(32648754647673077760\) | \([2]\) | \(7864320\) | \(2.7857\) | |
221760.cu2 | 221760lr3 | \([0, 0, 0, -5439468, 4633345712]\) | \(93137706732176569/5369647977540\) | \(1026155708580275159040\) | \([2]\) | \(7864320\) | \(2.7857\) | |
221760.cu3 | 221760lr2 | \([0, 0, 0, -1004268, -296822608]\) | \(586145095611769/140040608400\) | \(26762177026090598400\) | \([2, 2]\) | \(3932160\) | \(2.4391\) | |
221760.cu4 | 221760lr1 | \([0, 0, 0, 147732, -29097808]\) | \(1865864036231/2993760000\) | \(-572116445429760000\) | \([2]\) | \(1966080\) | \(2.0925\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 221760.cu have rank \(1\).
Complex multiplication
The elliptic curves in class 221760.cu do not have complex multiplication.Modular form 221760.2.a.cu
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.