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SageMath
E = EllipticCurve("da1")
E.isogeny_class()
Elliptic curves in class 61710.da
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
61710.da1 | 61710cz8 | \([1, 0, 0, -24349614385, 1462465543989737]\) | \(901247067798311192691198986281/552431869440\) | \(978666755056995840\) | \([2]\) | \(79626240\) | \(4.1597\) | |
61710.da2 | 61710cz7 | \([1, 0, 0, -1532073265, 22528378613225]\) | \(224494757451893010998773801/6152490825146276160000\) | \(10899512798686962140285760000\) | \([2]\) | \(79626240\) | \(4.1597\) | |
61710.da3 | 61710cz6 | \([1, 0, 0, -1521851185, 22850919992297]\) | \(220031146443748723000125481/172266701724057600\) | \(305180970372973205913600\) | \([2, 2]\) | \(39813120\) | \(3.8132\) | |
61710.da4 | 61710cz5 | \([1, 0, 0, -300673810, 2005244394572]\) | \(1696892787277117093383481/1440538624914939000\) | \(2552002046892934249779000\) | \([2]\) | \(26542080\) | \(3.6104\) | |
61710.da5 | 61710cz4 | \([1, 0, 0, -196913890, -1052215504900]\) | \(476646772170172569823801/5862293314453125000\) | \(10385410206445892578125000\) | \([2]\) | \(26542080\) | \(3.6104\) | |
61710.da6 | 61710cz3 | \([1, 0, 0, -94477105, 362070412265]\) | \(-52643812360427830814761/1504091705903677440\) | \(-2664590206602424709283840\) | \([4]\) | \(19906560\) | \(3.4666\) | |
61710.da7 | 61710cz2 | \([1, 0, 0, -22978810, 16337265572]\) | \(757443433548897303481/373234243041000000\) | \(661207228835957001000000\) | \([2, 2]\) | \(13271040\) | \(3.2639\) | |
61710.da8 | 61710cz1 | \([1, 0, 0, 5248070, 1958492900]\) | \(9023321954633914439/6156756739584000\) | \(-10907070126334170624000\) | \([4]\) | \(6635520\) | \(2.9173\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 61710.da have rank \(0\).
Complex multiplication
The elliptic curves in class 61710.da do not have complex multiplication.Modular form 61710.2.a.da
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 4 & 2 & 3 & 12 & 4 & 6 & 12 \\ 4 & 1 & 2 & 12 & 3 & 4 & 6 & 12 \\ 2 & 2 & 1 & 6 & 6 & 2 & 3 & 6 \\ 3 & 12 & 6 & 1 & 4 & 12 & 2 & 4 \\ 12 & 3 & 6 & 4 & 1 & 12 & 2 & 4 \\ 4 & 4 & 2 & 12 & 12 & 1 & 6 & 3 \\ 6 & 6 & 3 & 2 & 2 & 6 & 1 & 2 \\ 12 & 12 & 6 & 4 & 4 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.