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SageMath
E = EllipticCurve("bn1")
E.isogeny_class()
Elliptic curves in class 37026.bn
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 37026.bn1 | 37026bi6 | \([1, -1, 1, -430094039, -3433046186325]\) | \(6812873765474836663297/74052\) | \(95635786040388\) | \([2]\) | \(3932160\) | \(3.1862\) | |
| 37026.bn2 | 37026bi4 | \([1, -1, 1, -26880899, -53636217357]\) | \(1663303207415737537/5483698704\) | \(7082021227862812176\) | \([2, 2]\) | \(1966080\) | \(2.8396\) | |
| 37026.bn3 | 37026bi5 | \([1, -1, 1, -26510639, -55185829509]\) | \(-1595514095015181697/95635786040388\) | \(-123510554361298442331972\) | \([2]\) | \(3932160\) | \(3.1862\) | |
| 37026.bn4 | 37026bi2 | \([1, -1, 1, -1703219, -813444717]\) | \(423108074414017/23284318464\) | \(30070951476251279616\) | \([2, 2]\) | \(983040\) | \(2.4930\) | |
| 37026.bn5 | 37026bi1 | \([1, -1, 1, -309299, 50228115]\) | \(2533811507137/625016832\) | \(807189218613854208\) | \([4]\) | \(491520\) | \(2.1465\) | \(\Gamma_0(N)\)-optimal |
| 37026.bn6 | 37026bi3 | \([1, -1, 1, 1171741, -3282460365]\) | \(137763859017023/3683199928848\) | \(-4756734731530271069712\) | \([2]\) | \(1966080\) | \(2.8396\) |
Rank
sage: E.rank()
The elliptic curves in class 37026.bn have rank \(1\).
Complex multiplication
The elliptic curves in class 37026.bn do not have complex multiplication.Modular form 37026.2.a.bn
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
