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SageMath
E = EllipticCurve("bl1")
E.isogeny_class()
Elliptic curves in class 25410.bl
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 25410.bl1 | 25410bj8 | \([1, 0, 1, -232416803, 1363776582848]\) | \(783736670177727068275201/360150\) | \(638027694150\) | \([2]\) | \(2621440\) | \(2.9934\) | |
| 25410.bl2 | 25410bj6 | \([1, 0, 1, -14526053, 21308093948]\) | \(191342053882402567201/129708022500\) | \(229785674048122500\) | \([2, 2]\) | \(1310720\) | \(2.6468\) | |
| 25410.bl3 | 25410bj7 | \([1, 0, 1, -14435303, 21587495048]\) | \(-187778242790732059201/4984939585440150\) | \(-8831124556921937574150\) | \([2]\) | \(2621440\) | \(2.9934\) | |
| 25410.bl4 | 25410bj4 | \([1, 0, 1, -1823473, -947648044]\) | \(378499465220294881/120530818800\) | \(213527697884146800\) | \([2]\) | \(655360\) | \(2.3002\) | |
| 25410.bl5 | 25410bj3 | \([1, 0, 1, -913553, 328508948]\) | \(47595748626367201/1215506250000\) | \(2153343467756250000\) | \([2, 2]\) | \(655360\) | \(2.3002\) | |
| 25410.bl6 | 25410bj2 | \([1, 0, 1, -129473, -10527244]\) | \(135487869158881/51438240000\) | \(91125979892640000\) | \([2, 2]\) | \(327680\) | \(1.9537\) | |
| 25410.bl7 | 25410bj1 | \([1, 0, 1, 25407, -1172492]\) | \(1023887723039/928972800\) | \(-1645731982540800\) | \([2]\) | \(163840\) | \(1.6071\) | \(\Gamma_0(N)\)-optimal |
| 25410.bl8 | 25410bj5 | \([1, 0, 1, 153667, 1050376556]\) | \(226523624554079/269165039062500\) | \(-476842285766601562500\) | \([2]\) | \(1310720\) | \(2.6468\) |
Rank
sage: E.rank()
The elliptic curves in class 25410.bl have rank \(1\).
Complex multiplication
The elliptic curves in class 25410.bl do not have complex multiplication.Modular form 25410.2.a.bl
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 16 & 4 & 8 & 16 & 8 \\ 2 & 1 & 2 & 8 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 16 & 4 & 8 & 16 & 8 \\ 16 & 8 & 16 & 1 & 4 & 2 & 4 & 8 \\ 4 & 2 & 4 & 4 & 1 & 2 & 4 & 2 \\ 8 & 4 & 8 & 2 & 2 & 1 & 2 & 4 \\ 16 & 8 & 16 & 4 & 4 & 2 & 1 & 8 \\ 8 & 4 & 8 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
