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SageMath
E = EllipticCurve("d1")
E.isogeny_class()
Elliptic curves in class 1785.d
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 1785.d1 | 1785m5 | \([1, 0, 0, -23170, -1359403]\) | \(1375634265228629281/24990412335\) | \(24990412335\) | \([2]\) | \(4096\) | \(1.1213\) | |
| 1785.d2 | 1785m4 | \([1, 0, 0, -5725, 166250]\) | \(20751759537944401/418359375\) | \(418359375\) | \([4]\) | \(2048\) | \(0.77468\) | |
| 1785.d3 | 1785m3 | \([1, 0, 0, -1495, -19888]\) | \(369543396484081/45120132225\) | \(45120132225\) | \([2, 2]\) | \(2048\) | \(0.77468\) | |
| 1785.d4 | 1785m2 | \([1, 0, 0, -370, 2387]\) | \(5602762882081/716900625\) | \(716900625\) | \([2, 4]\) | \(1024\) | \(0.42810\) | |
| 1785.d5 | 1785m1 | \([1, 0, 0, 35, 200]\) | \(4733169839/19518975\) | \(-19518975\) | \([4]\) | \(512\) | \(0.081530\) | \(\Gamma_0(N)\)-optimal |
| 1785.d6 | 1785m6 | \([1, 0, 0, 2180, -101473]\) | \(1145725929069119/5127181719135\) | \(-5127181719135\) | \([2]\) | \(4096\) | \(1.1213\) |
Rank
sage: E.rank()
The elliptic curves in class 1785.d have rank \(0\).
Complex multiplication
The elliptic curves in class 1785.d do not have complex multiplication.Modular form 1785.2.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.
