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SageMath
E = EllipticCurve("bm1")
E.isogeny_class()
Elliptic curves in class 86190bm
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 86190.bm7 | 86190bm1 | \([1, 0, 1, -354248678, 2471124782648]\) | \(1018563973439611524445729/42904970360310988800\) | \(207094097079882323538739200\) | \([2]\) | \(46448640\) | \(3.8142\) | \(\Gamma_0(N)\)-optimal |
| 86190.bm6 | 86190bm2 | \([1, 0, 1, -939177958, -7792278335944]\) | \(18980483520595353274840609/5549773448629762560000\) | \(26787696429807175592471040000\) | \([2, 2]\) | \(92897280\) | \(4.1608\) | |
| 86190.bm5 | 86190bm3 | \([1, 0, 1, -4358980838, -110049775142344]\) | \(1897660325010178513043539489/14258428094958372000000\) | \(68822709054597924594948000000\) | \([2]\) | \(139345920\) | \(4.3635\) | |
| 86190.bm8 | 86190bm4 | \([1, 0, 1, 2514857562, -51791164403912]\) | \(364421318680576777174674911/450962301637624725000000\) | \(-2176708896205201761252525000000\) | \([4]\) | \(185794560\) | \(4.5074\) | |
| 86190.bm4 | 86190bm5 | \([1, 0, 1, -13752081958, -620648852140744]\) | \(59589391972023341137821784609/8834417507562311995200\) | \(42642045935259335599239316800\) | \([2]\) | \(185794560\) | \(4.5074\) | |
| 86190.bm2 | 86190bm6 | \([1, 0, 1, -69617438518, -7070099217161992]\) | \(7730680381889320597382223137569/441370202660156250000\) | \(2130409666531866128906250000\) | \([2, 2]\) | \(278691840\) | \(4.7101\) | |
| 86190.bm3 | 86190bm7 | \([1, 0, 1, -69491198898, -7097017240838744]\) | \(-7688701694683937879808871873249/58423707246780395507812500\) | \(-282000075952124834060668945312500\) | \([4]\) | \(557383680\) | \(5.0567\) | |
| 86190.bm1 | 86190bm8 | \([1, 0, 1, -1113879001018, -452486084448911992]\) | \(31664865542564944883878115208137569/103216295812500\) | \(498205345574437312500\) | \([2]\) | \(557383680\) | \(5.0567\) |
Rank
sage: E.rank()
The elliptic curves in class 86190bm have rank \(0\).
Complex multiplication
The elliptic curves in class 86190bm do not have complex multiplication.Modular form 86190.2.a.bm
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 Cremona numbering.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
