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SageMath
E = EllipticCurve("ej1")
E.isogeny_class()
Elliptic curves in class 139650.ej
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 139650.ej1 | 139650he4 | \([1, 0, 1, -567458001, -3322369451852]\) | \(10993009831928446009969/3767761230468750000\) | \(6926145953178405761718750000\) | \([2]\) | \(149299200\) | \(4.0440\) | |
| 139650.ej2 | 139650he2 | \([1, 0, 1, -508364001, -4411785863852]\) | \(7903870428425797297009/886464000000\) | \(1629556299000000000000\) | \([2]\) | \(49766400\) | \(3.4947\) | |
| 139650.ej3 | 139650he1 | \([1, 0, 1, -31692001, -69303943852]\) | \(-1914980734749238129/20440940544000\) | \(-37575878344704000000000\) | \([2]\) | \(24883200\) | \(3.1482\) | \(\Gamma_0(N)\)-optimal |
| 139650.ej4 | 139650he3 | \([1, 0, 1, 104723999, -360735559852]\) | \(69096190760262356111/70568821500000000\) | \(-129724238760210937500000000\) | \([2]\) | \(74649600\) | \(3.6975\) |
Rank
sage: E.rank()
The elliptic curves in class 139650.ej have rank \(1\).
Complex multiplication
The elliptic curves in class 139650.ej do not have complex multiplication.Modular form 139650.2.a.ej
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
