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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 42483.d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 42483.d1 | 42483v6 | \([1, 0, 0, -11102519, 14238104670]\) | \(53297461115137/147\) | \(417444845726307\) | \([2]\) | \(884736\) | \(2.4638\) | |
| 42483.d2 | 42483v4 | \([1, 0, 0, -694184, 222240759]\) | \(13027640977/21609\) | \(61364392321767129\) | \([2, 2]\) | \(442368\) | \(2.1172\) | |
| 42483.d3 | 42483v3 | \([1, 0, 0, -552574, -157189075]\) | \(6570725617/45927\) | \(130421696800490487\) | \([2]\) | \(442368\) | \(2.1172\) | |
| 42483.d4 | 42483v5 | \([1, 0, 0, -481769, 360862788]\) | \(-4354703137/17294403\) | \(-49111968654854292243\) | \([2]\) | \(884736\) | \(2.4638\) | |
| 42483.d5 | 42483v2 | \([1, 0, 0, -56939, 1116744]\) | \(7189057/3969\) | \(11271010834610289\) | \([2, 2]\) | \(221184\) | \(1.7706\) | |
| 42483.d6 | 42483v1 | \([1, 0, 0, 13866, 139635]\) | \(103823/63\) | \(-178904933882703\) | \([2]\) | \(110592\) | \(1.4240\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 42483.d have rank \(1\).
Complex multiplication
The elliptic curves in class 42483.d do not have complex multiplication.Modular form 42483.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 8 & 4 & 4 & 8 \\ 2 & 1 & 4 & 2 & 2 & 4 \\ 8 & 4 & 1 & 8 & 2 & 4 \\ 4 & 2 & 8 & 1 & 4 & 8 \\ 4 & 2 & 2 & 4 & 1 & 2 \\ 8 & 4 & 4 & 8 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
