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SageMath
E = EllipticCurve("cp1")
E.isogeny_class()
Elliptic curves in class 164730.cp
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 164730.cp1 | 164730o4 | \([1, 1, 1, -493196135, -2605512415363]\) | \(549653727492794875187089/196605747070312500000\) | \(4745584785706215820312500000\) | \([2]\) | \(141557760\) | \(4.0115\) | |
| 164730.cp2 | 164730o2 | \([1, 1, 1, -209421255, 1136797192125]\) | \(42081620701292477662609/1220273715600000000\) | \(29454441009181376400000000\) | \([2, 2]\) | \(70778880\) | \(3.6649\) | |
| 164730.cp3 | 164730o1 | \([1, 1, 1, -207941575, 1154056771517]\) | \(41195916697879355491729/36197498880000\) | \(873719626843422720000\) | \([4]\) | \(35389440\) | \(3.3183\) | \(\Gamma_0(N)\)-optimal |
| 164730.cp4 | 164730o3 | \([1, 1, 1, 50678745, 3774523312125]\) | \(596358945261507937391/255327150374524980000\) | \(-6162976709738472546973620000\) | \([2]\) | \(141557760\) | \(4.0115\) |
Rank
sage: E.rank()
The elliptic curves in class 164730.cp have rank \(1\).
Complex multiplication
The elliptic curves in class 164730.cp do not have complex multiplication.Modular form 164730.2.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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
