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SageMath
E = EllipticCurve("cp1")
E.isogeny_class()
Elliptic curves in class 59200.cp
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 59200.cp1 | 59200x3 | \([0, 1, 0, -187333, -31270787]\) | \(727057727488000/37\) | \(37000000\) | \([]\) | \(124416\) | \(1.3734\) | |
| 59200.cp2 | 59200x2 | \([0, 1, 0, -2333, -42787]\) | \(1404928000/50653\) | \(50653000000\) | \([]\) | \(41472\) | \(0.82407\) | |
| 59200.cp3 | 59200x1 | \([0, 1, 0, -333, 2213]\) | \(4096000/37\) | \(37000000\) | \([]\) | \(13824\) | \(0.27476\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 59200.cp have rank \(2\).
Complex multiplication
The elliptic curves in class 59200.cp do not have complex multiplication.Modular form 59200.2.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.
\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
