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SageMath
E = EllipticCurve("c1")
E.isogeny_class()
Elliptic curves in class 2205.c
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
2205.c1 | 2205d2 | \([1, -1, 1, -167, 634]\) | \(55306341/15625\) | \(144703125\) | \([2]\) | \(768\) | \(0.27272\) | |
2205.c2 | 2205d1 | \([1, -1, 1, -62, -164]\) | \(2803221/125\) | \(1157625\) | \([2]\) | \(384\) | \(-0.073855\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 2205.c have rank \(1\).
Complex multiplication
The elliptic curves in class 2205.c do not have complex multiplication.Modular form 2205.2.a.c
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.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.