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SageMath
E = EllipticCurve("ej1")
E.isogeny_class()
Elliptic curves in class 288990ej
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 288990.ej3 | 288990ej1 | \([1, -1, 1, -47183, -3909369]\) | \(3301293169/22800\) | \(80227357750800\) | \([2]\) | \(1179648\) | \(1.5016\) | \(\Gamma_0(N)\)-optimal |
| 288990.ej2 | 288990ej2 | \([1, -1, 1, -77603, 1773087]\) | \(14688124849/8122500\) | \(28580996198722500\) | \([2, 2]\) | \(2359296\) | \(1.8481\) | |
| 288990.ej1 | 288990ej3 | \([1, -1, 1, -944573, 353069331]\) | \(26487576322129/44531250\) | \(156694058107031250\) | \([2]\) | \(4718592\) | \(2.1947\) | |
| 288990.ej4 | 288990ej4 | \([1, -1, 1, 302647, 13788987]\) | \(871257511151/527800050\) | \(-1857193132992988050\) | \([2]\) | \(4718592\) | \(2.1947\) |
Rank
sage: E.rank()
The elliptic curves in class 288990ej have rank \(0\).
Complex multiplication
The elliptic curves in class 288990ej do not have complex multiplication.Modular form 288990.2.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 Cremona 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 Cremona labels.
