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SageMath
E = EllipticCurve("du1")
E.isogeny_class()
Elliptic curves in class 53550.du
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
53550.du1 | 53550eb6 | \([1, -1, 1, -392080505, 2988303444497]\) | \(585196747116290735872321/836876053125000\) | \(9532541292626953125000\) | \([2]\) | \(14155776\) | \(3.4887\) | |
53550.du2 | 53550eb4 | \([1, -1, 1, -56839505, -164898629503]\) | \(1782900110862842086081/328139630024640\) | \(3737715473249415000000\) | \([2]\) | \(7077888\) | \(3.1421\) | |
53550.du3 | 53550eb3 | \([1, -1, 1, -24727505, 45805914497]\) | \(146796951366228945601/5397929064360000\) | \(61485785748725625000000\) | \([2, 2]\) | \(7077888\) | \(3.1421\) | |
53550.du4 | 53550eb2 | \([1, -1, 1, -3919505, -2010869503]\) | \(584614687782041281/184812061593600\) | \(2105124889089600000000\) | \([2, 2]\) | \(3538944\) | \(2.7955\) | |
53550.du5 | 53550eb1 | \([1, -1, 1, 688495, -213749503]\) | \(3168685387909439/3563732336640\) | \(-40593138647040000000\) | \([4]\) | \(1769472\) | \(2.4490\) | \(\Gamma_0(N)\)-optimal |
53550.du6 | 53550eb5 | \([1, -1, 1, 9697495, 163332864497]\) | \(8854313460877886399/1016927675429790600\) | \(-11583441802942458553125000\) | \([2]\) | \(14155776\) | \(3.4887\) |
Rank
sage: E.rank()
The elliptic curves in class 53550.du have rank \(1\).
Complex multiplication
The elliptic curves in class 53550.du do not have complex multiplication.Modular form 53550.2.a.du
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.
\(\left(\begin{array}{rrrrrr} 1 & 8 & 2 & 4 & 8 & 4 \\ 8 & 1 & 4 & 2 & 4 & 8 \\ 2 & 4 & 1 & 2 & 4 & 2 \\ 4 & 2 & 2 & 1 & 2 & 4 \\ 8 & 4 & 4 & 2 & 1 & 8 \\ 4 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.