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SageMath
E = EllipticCurve("ic1")
E.isogeny_class()
Elliptic curves in class 139650.ic
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 139650.ic1 | 139650bt4 | \([1, 0, 0, -108948463, -422413566583]\) | \(77799851782095807001/3092322318750000\) | \(5684509819978417968750000\) | \([2]\) | \(28311552\) | \(3.5166\) | |
| 139650.ic2 | 139650bt2 | \([1, 0, 0, -17710463, 19817019417]\) | \(334199035754662681/101099003040000\) | \(185846822010202500000000\) | \([2, 2]\) | \(14155776\) | \(3.1700\) | |
| 139650.ic3 | 139650bt1 | \([1, 0, 0, -16142463, 24958491417]\) | \(253060782505556761/41184460800\) | \(75707978572800000000\) | \([2]\) | \(7077888\) | \(2.8235\) | \(\Gamma_0(N)\)-optimal |
| 139650.ic4 | 139650bt3 | \([1, 0, 0, 48439537, 132999669417]\) | \(6837784281928633319/8113766016106800\) | \(-14915257156702326768750000\) | \([2]\) | \(28311552\) | \(3.5166\) |
Rank
sage: E.rank()
The elliptic curves in class 139650.ic have rank \(1\).
Complex multiplication
The elliptic curves in class 139650.ic do not have complex multiplication.Modular form 139650.2.a.ic
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.
