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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 192.d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 192.d1 | 192c5 | \([0, 1, 0, -1537, -23713]\) | \(3065617154/9\) | \(1179648\) | \([2]\) | \(64\) | \(0.39437\) | |
| 192.d2 | 192c4 | \([0, 1, 0, -257, 1503]\) | \(28756228/3\) | \(196608\) | \([2]\) | \(32\) | \(0.047795\) | |
| 192.d3 | 192c3 | \([0, 1, 0, -97, -385]\) | \(1556068/81\) | \(5308416\) | \([2, 2]\) | \(32\) | \(0.047795\) | |
| 192.d4 | 192c2 | \([0, 1, 0, -17, 15]\) | \(35152/9\) | \(147456\) | \([2, 2]\) | \(16\) | \(-0.29878\) | |
| 192.d5 | 192c1 | \([0, 1, 0, 3, 3]\) | \(2048/3\) | \(-3072\) | \([2]\) | \(8\) | \(-0.64535\) | \(\Gamma_0(N)\)-optimal |
| 192.d6 | 192c6 | \([0, 1, 0, 63, -1377]\) | \(207646/6561\) | \(-859963392\) | \([4]\) | \(64\) | \(0.39437\) |
Rank
sage: E.rank()
The elliptic curves in class 192.d have rank \(0\).
Complex multiplication
The elliptic curves in class 192.d do not have complex multiplication.Modular form 192.2.a.d
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 8 & 4 \\ 8 & 1 & 4 & 2 & 4 & 8 \\ 2 & 4 & 1 & 2 & 4 & 2 \\ 4 & 2 & 2 & 1 & 2 & 4 \\ 8 & 4 & 4 & 2 & 1 & 8 \\ 4 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
