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SageMath
E = EllipticCurve("e1")
E.isogeny_class()
Elliptic curves in class 8050.e
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 8050.e1 | 8050e3 | \([1, -1, 0, -914942, 337080216]\) | \(5421065386069310769/1919709260\) | \(29995457187500\) | \([4]\) | \(73728\) | \(1.9404\) | |
| 8050.e2 | 8050e2 | \([1, -1, 0, -57442, 5227716]\) | \(1341518286067569/24894528400\) | \(388977006250000\) | \([2, 2]\) | \(36864\) | \(1.5939\) | |
| 8050.e3 | 8050e1 | \([1, -1, 0, -7442, -122284]\) | \(2917464019569/1262240000\) | \(19722500000000\) | \([2]\) | \(18432\) | \(1.2473\) | \(\Gamma_0(N)\)-optimal |
| 8050.e4 | 8050e4 | \([1, -1, 0, 58, 15175216]\) | \(1367631/6366992112460\) | \(-99484251757187500\) | \([2]\) | \(73728\) | \(1.9404\) |
Rank
sage: E.rank()
The elliptic curves in class 8050.e have rank \(0\).
Complex multiplication
The elliptic curves in class 8050.e do not have complex multiplication.Modular form 8050.2.a.e
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.
