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SageMath
E = EllipticCurve("dl1")
E.isogeny_class()
Elliptic curves in class 287490.dl
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 287490.dl1 | 287490dl8 | \([1, 0, 0, -480837321, -4058351458599]\) | \(4791901410190533590281/41160000\) | \(105605298994440000\) | \([2]\) | \(55738368\) | \(3.3077\) | |
| 287490.dl2 | 287490dl6 | \([1, 0, 0, -30053001, -63410657895]\) | \(1169975873419524361/108425318400\) | \(278189702823113625600\) | \([2, 2]\) | \(27869184\) | \(2.9612\) | |
| 287490.dl3 | 287490dl7 | \([1, 0, 0, -27862601, -73045351335]\) | \(-932348627918877961/358766164249920\) | \(-920495822271651420137280\) | \([2]\) | \(55738368\) | \(3.3077\) | |
| 287490.dl4 | 287490dl5 | \([1, 0, 0, -5965446, -5509954224]\) | \(9150443179640281/184570312500\) | \(473556925098632812500\) | \([2]\) | \(18579456\) | \(2.7584\) | |
| 287490.dl5 | 287490dl3 | \([1, 0, 0, -2015881, -837413479]\) | \(353108405631241/86318776320\) | \(221470363996787834880\) | \([2]\) | \(13934592\) | \(2.6146\) | |
| 287490.dl6 | 287490dl2 | \([1, 0, 0, -790626, 141984180]\) | \(21302308926361/8930250000\) | \(22912578263972250000\) | \([2, 2]\) | \(9289728\) | \(2.4118\) | |
| 287490.dl7 | 287490dl1 | \([1, 0, 0, -681106, 216216836]\) | \(13619385906841/6048000\) | \(15517513321632000\) | \([2]\) | \(4644864\) | \(2.0653\) | \(\Gamma_0(N)\)-optimal |
| 287490.dl8 | 287490dl4 | \([1, 0, 0, 2631874, 1043470680]\) | \(785793873833639/637994920500\) | \(-1636920416334705484500\) | \([2]\) | \(18579456\) | \(2.7584\) |
Rank
sage: E.rank()
The elliptic curves in class 287490.dl have rank \(0\).
Complex multiplication
The elliptic curves in class 287490.dl do not have complex multiplication.Modular form 287490.2.a.dl
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 3 & 4 & 6 & 12 & 12 \\ 2 & 1 & 2 & 6 & 2 & 3 & 6 & 6 \\ 4 & 2 & 1 & 12 & 4 & 6 & 12 & 3 \\ 3 & 6 & 12 & 1 & 12 & 2 & 4 & 4 \\ 4 & 2 & 4 & 12 & 1 & 6 & 3 & 12 \\ 6 & 3 & 6 & 2 & 6 & 1 & 2 & 2 \\ 12 & 6 & 12 & 4 & 3 & 2 & 1 & 4 \\ 12 & 6 & 3 & 4 & 12 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
