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SageMath
E = EllipticCurve("cp1")
E.isogeny_class()
Elliptic curves in class 55770.cp
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
55770.cp1 | 55770cs4 | \([1, 0, 0, -383895756, -2895162061104]\) | \(1296294060988412126189641/647824320\) | \(3126924258194880\) | \([2]\) | \(6967296\) | \(3.2094\) | |
55770.cp2 | 55770cs3 | \([1, 0, 0, -23993356, -45238916464]\) | \(-316472948332146183241/7074906009600\) | \(-34149220001291366400\) | \([2]\) | \(3483648\) | \(2.8628\) | |
55770.cp3 | 55770cs2 | \([1, 0, 0, -4748481, -3955879539]\) | \(2453170411237305241/19353090685500\) | \(93413672298587569500\) | \([2]\) | \(2322432\) | \(2.6600\) | |
55770.cp4 | 55770cs1 | \([1, 0, 0, -100981, -142141039]\) | \(-23592983745241/1794399750000\) | \(-8661224862897750000\) | \([2]\) | \(1161216\) | \(2.3135\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 55770.cp have rank \(0\).
Complex multiplication
The elliptic curves in class 55770.cp do not have complex multiplication.Modular form 55770.2.a.cp
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.