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SageMath
E = EllipticCurve("bo1")
E.isogeny_class()
Elliptic curves in class 27225bo
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
27225.r3 | 27225bo1 | \([1, -1, 1, -177530, -28504528]\) | \(30664297/297\) | \(5993218543640625\) | \([2]\) | \(184320\) | \(1.8472\) | \(\Gamma_0(N)\)-optimal |
27225.r2 | 27225bo2 | \([1, -1, 1, -313655, 21317222]\) | \(169112377/88209\) | \(1779985907461265625\) | \([2, 2]\) | \(368640\) | \(2.1938\) | |
27225.r4 | 27225bo3 | \([1, -1, 1, 1183720, 165065222]\) | \(9090072503/5845851\) | \(-117964520594478421875\) | \([2]\) | \(737280\) | \(2.5403\) | |
27225.r1 | 27225bo4 | \([1, -1, 1, -3989030, 3064527722]\) | \(347873904937/395307\) | \(7976973881585671875\) | \([2]\) | \(737280\) | \(2.5403\) |
Rank
sage: E.rank()
The elliptic curves in class 27225bo have rank \(1\).
Complex multiplication
The elliptic curves in class 27225bo do not have complex multiplication.Modular form 27225.2.a.bo
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.