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SageMath
E = EllipticCurve("n1")
E.isogeny_class()
Elliptic curves in class 19890.n
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
19890.n1 | 19890r7 | \([1, -1, 0, -59319000054, 5560834936210128]\) | \(31664865542564944883878115208137569/103216295812500\) | \(75244679647312500\) | \([6]\) | \(26542080\) | \(4.3235\) | |
19890.n2 | 19890r6 | \([1, -1, 0, -3707437554, 86888738522628]\) | \(7730680381889320597382223137569/441370202660156250000\) | \(321758877739253906250000\) | \([2, 6]\) | \(13271040\) | \(3.9769\) | |
19890.n3 | 19890r8 | \([1, -1, 0, -3700714734, 87219545637240]\) | \(-7688701694683937879808871873249/58423707246780395507812500\) | \(-42590882582902908325195312500\) | \([6]\) | \(26542080\) | \(4.3235\) | |
19890.n4 | 19890r4 | \([1, -1, 0, -732359394, 7627624175700]\) | \(59589391972023341137821784609/8834417507562311995200\) | \(6440290363012925444500800\) | \([2]\) | \(8847360\) | \(3.7742\) | |
19890.n5 | 19890r3 | \([1, -1, 0, -232135074, 1352508703380]\) | \(1897660325010178513043539489/14258428094958372000000\) | \(10394394081224653188000000\) | \([6]\) | \(6635520\) | \(3.6304\) | |
19890.n6 | 19890r2 | \([1, -1, 0, -50015394, 95774634900]\) | \(18980483520595353274840609/5549773448629762560000\) | \(4045784844051096906240000\) | \([2, 2]\) | \(4423680\) | \(3.4276\) | |
19890.n7 | 19890r1 | \([1, -1, 0, -18865314, -30364499052]\) | \(1018563973439611524445729/42904970360310988800\) | \(31277723392666710835200\) | \([2]\) | \(2211840\) | \(3.0811\) | \(\Gamma_0(N)\)-optimal |
19890.n8 | 19890r5 | \([1, -1, 0, 133927326, 636455866068]\) | \(364421318680576777174674911/450962301637624725000000\) | \(-328751517893828424525000000\) | \([2]\) | \(8847360\) | \(3.7742\) |
Rank
sage: E.rank()
The elliptic curves in class 19890.n have rank \(0\).
Complex multiplication
The elliptic curves in class 19890.n do not have complex multiplication.Modular form 19890.2.a.n
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.