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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 133308.j
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 133308.j1 | 133308i4 | \([0, 0, 0, -8704695, -9885030482]\) | \(2640279346000/3087\) | \(85284702574348032\) | \([2]\) | \(3421440\) | \(2.5319\) | |
| 133308.j2 | 133308i3 | \([0, 0, 0, -539580, -157112471]\) | \(-10061824000/352947\) | \(-609430270479195312\) | \([2]\) | \(1710720\) | \(2.1853\) | |
| 133308.j3 | 133308i2 | \([0, 0, 0, -134895, -6107834]\) | \(9826000/5103\) | \(140980834867799808\) | \([2]\) | \(1140480\) | \(1.9826\) | |
| 133308.j4 | 133308i1 | \([0, 0, 0, 31740, -742187]\) | \(2048000/1323\) | \(-2284411676098608\) | \([2]\) | \(570240\) | \(1.6360\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 133308.j have rank \(0\).
Complex multiplication
The elliptic curves in class 133308.j do not have complex multiplication.Modular form 133308.2.a.j
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
