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SageMath
E = EllipticCurve("r1")
E.isogeny_class()
Elliptic curves in class 328560.r
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
328560.r1 | 328560r3 | \([0, -1, 0, -7019323440, 46602817329600]\) | \(3639478711331685826729/2016912141902025000\) | \(21196164084165759111488409600000\) | \([2]\) | \(756449280\) | \(4.7014\) | |
328560.r2 | 328560r2 | \([0, -1, 0, -4281323440, -107187357070400]\) | \(825824067562227826729/5613755625000000\) | \(58996166904769743360000000000\) | \([2, 2]\) | \(378224640\) | \(4.3548\) | |
328560.r3 | 328560r1 | \([0, -1, 0, -4274314160, -107557822751808]\) | \(821774646379511057449/38361600000\) | \(403150316386281062400000\) | \([2]\) | \(189112320\) | \(4.0082\) | \(\Gamma_0(N)\)-optimal |
328560.r4 | 328560r4 | \([0, -1, 0, -1655471920, -237267840008768]\) | \(-47744008200656797609/2286529541015625000\) | \(-24029631399290625000000000000000\) | \([4]\) | \(756449280\) | \(4.7014\) |
Rank
sage: E.rank()
The elliptic curves in class 328560.r have rank \(1\).
Complex multiplication
The elliptic curves in class 328560.r do not have complex multiplication.Modular form 328560.2.a.r
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.