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SageMath
E = EllipticCurve("z1")
E.isogeny_class()
Elliptic curves in class 139230z
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 139230.df7 | 139230z1 | \([1, -1, 1, -571253, 118995581]\) | \(28280100765151839241/7994847656250000\) | \(5828243941406250000\) | \([2]\) | \(2433024\) | \(2.3076\) | \(\Gamma_0(N)\)-optimal |
| 139230.df6 | 139230z2 | \([1, -1, 1, -3383753, -2300879419]\) | \(5877491705974396839241/261806444735062500\) | \(190856898211860562500\) | \([2, 2]\) | \(4866048\) | \(2.6542\) | |
| 139230.df4 | 139230z3 | \([1, -1, 1, -42505628, 106674596831]\) | \(11650256451486052494789241/580277967360000\) | \(423022638205440000\) | \([6]\) | \(7299072\) | \(2.8569\) | |
| 139230.df8 | 139230z4 | \([1, -1, 1, 1779997, -8701863919]\) | \(855567391070976980759/45363085180055574750\) | \(-33069689096260513992750\) | \([2]\) | \(9732096\) | \(3.0008\) | |
| 139230.df2 | 139230z5 | \([1, -1, 1, -53547503, -150805644919]\) | \(23292378980986805290659241/49479832772574750\) | \(36070798091206992750\) | \([2]\) | \(9732096\) | \(3.0008\) | |
| 139230.df3 | 139230z6 | \([1, -1, 1, -42577628, 106295128031]\) | \(11709559667189768059461241/82207646338733697600\) | \(59929374180936865550400\) | \([2, 6]\) | \(14598144\) | \(3.2035\) | |
| 139230.df5 | 139230z7 | \([1, -1, 1, -16047428, 237205746911]\) | \(-626920492174472718626041/32979221374608565962360\) | \(-24041852382089644586560440\) | \([6]\) | \(29196288\) | \(3.5501\) | |
| 139230.df1 | 139230z8 | \([1, -1, 1, -70259828, -48902358049]\) | \(52615951054626272117608441/29030877531795041917560\) | \(21163509720678585557901240\) | \([6]\) | \(29196288\) | \(3.5501\) |
Rank
sage: E.rank()
The elliptic curves in class 139230z have rank \(0\).
Complex multiplication
The elliptic curves in class 139230z do not have complex multiplication.Modular form 139230.2.a.z
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 Cremona numbering.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
