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SageMath
E = EllipticCurve("ba1")
E.isogeny_class()
Elliptic curves in class 296010.ba
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 296010.ba1 | 296010ba8 | \([1, -1, 0, -897768729, -10321037288397]\) | \(109771501864515940852538932369/397208046486476323205250\) | \(289564665888641239616627250\) | \([6]\) | \(233570304\) | \(3.9386\) | |
| 296010.ba2 | 296010ba5 | \([1, -1, 0, -896987619, -10339948316115]\) | \(109485228680228758274973692209/5229455928840\) | \(3812273372124360\) | \([2]\) | \(77856768\) | \(3.3893\) | |
| 296010.ba3 | 296010ba6 | \([1, -1, 0, -82132479, 3449491353]\) | \(84050631486759338471152369/48633444181761930562500\) | \(35453780808504447380062500\) | \([2, 6]\) | \(116785152\) | \(3.5920\) | |
| 296010.ba4 | 296010ba3 | \([1, -1, 0, -56819979, 164391428853]\) | \(27829114954005748466152369/93036058628906250000\) | \(67823286740472656250000\) | \([6]\) | \(58392576\) | \(3.2454\) | |
| 296010.ba5 | 296010ba2 | \([1, -1, 0, -56061819, -161550618075]\) | \(26729925166733144545311409/184733644634510400\) | \(134670826938558081600\) | \([2, 2]\) | \(38928384\) | \(3.0427\) | |
| 296010.ba6 | 296010ba4 | \([1, -1, 0, -54944019, -168302353635]\) | \(-25162712714293505984578609/2226574457229407468040\) | \(-1623172779320238044201160\) | \([2]\) | \(77856768\) | \(3.3893\) | |
| 296010.ba7 | 296010ba1 | \([1, -1, 0, -3573819, -2417499675]\) | \(6924587578730936223409/541432430922240000\) | \(394704242142312960000\) | \([2]\) | \(19464192\) | \(2.6961\) | \(\Gamma_0(N)\)-optimal |
| 296010.ba8 | 296010ba7 | \([1, -1, 0, 328503771, 27348521103]\) | \(5377956202510641533116627631/3112689063355048990205250\) | \(-2269150327185830713859627250\) | \([6]\) | \(233570304\) | \(3.9386\) |
Rank
sage: E.rank()
The elliptic curves in class 296010.ba have rank \(1\).
Complex multiplication
The elliptic curves in class 296010.ba do not have complex multiplication.Modular form 296010.2.a.ba
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 3 & 2 & 4 & 6 & 12 & 12 & 4 \\ 3 & 1 & 6 & 12 & 2 & 4 & 4 & 12 \\ 2 & 6 & 1 & 2 & 3 & 6 & 6 & 2 \\ 4 & 12 & 2 & 1 & 6 & 12 & 3 & 4 \\ 6 & 2 & 3 & 6 & 1 & 2 & 2 & 6 \\ 12 & 4 & 6 & 12 & 2 & 1 & 4 & 3 \\ 12 & 4 & 6 & 3 & 2 & 4 & 1 & 12 \\ 4 & 12 & 2 & 4 & 6 & 3 & 12 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
