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SageMath
E = EllipticCurve("bq1")
E.isogeny_class()
Elliptic curves in class 26775bq
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 26775.p5 | 26775bq1 | \([1, -1, 1, 7870, -667128]\) | \(4733169839/19518975\) | \(-222333324609375\) | \([2]\) | \(98304\) | \(1.4356\) | \(\Gamma_0(N)\)-optimal |
| 26775.p4 | 26775bq2 | \([1, -1, 1, -83255, -8139378]\) | \(5602762882081/716900625\) | \(8165946181640625\) | \([2, 2]\) | \(196608\) | \(1.7821\) | |
| 26775.p3 | 26775bq3 | \([1, -1, 1, -336380, 66785622]\) | \(369543396484081/45120132225\) | \(513946506125390625\) | \([2, 2]\) | \(393216\) | \(2.1287\) | |
| 26775.p2 | 26775bq4 | \([1, -1, 1, -1288130, -562381878]\) | \(20751759537944401/418359375\) | \(4765374755859375\) | \([2]\) | \(393216\) | \(2.1287\) | |
| 26775.p6 | 26775bq5 | \([1, -1, 1, 490495, 342961872]\) | \(1145725929069119/5127181719135\) | \(-58401804269522109375\) | \([2]\) | \(786432\) | \(2.4753\) | |
| 26775.p1 | 26775bq6 | \([1, -1, 1, -5213255, 4582771872]\) | \(1375634265228629281/24990412335\) | \(284656415503359375\) | \([4]\) | \(786432\) | \(2.4753\) |
Rank
sage: E.rank()
The elliptic curves in class 26775bq have rank \(2\).
Complex multiplication
The elliptic curves in class 26775bq do not have complex multiplication.Modular form 26775.2.a.bq
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 Cremona numbering.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 2 & 2 \\ 4 & 2 & 4 & 1 & 8 & 8 \\ 8 & 4 & 2 & 8 & 1 & 4 \\ 8 & 4 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
