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SageMath
E = EllipticCurve("eq1")
E.isogeny_class()
Elliptic curves in class 177600.eq
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
177600.eq1 | 177600em3 | \([0, -1, 0, -512733633, 920165335137]\) | \(3639478711331685826729/2016912141902025000\) | \(8261272133230694400000000000\) | \([2]\) | \(106168320\) | \(4.0472\) | |
177600.eq2 | 177600em2 | \([0, -1, 0, -312733633, -2116034664863]\) | \(825824067562227826729/5613755625000000\) | \(22993943040000000000000000\) | \([2, 2]\) | \(53084160\) | \(3.7007\) | |
177600.eq3 | 177600em1 | \([0, -1, 0, -312221633, -2123348584863]\) | \(821774646379511057449/38361600000\) | \(157129113600000000000\) | \([2]\) | \(26542080\) | \(3.3541\) | \(\Gamma_0(N)\)-optimal |
177600.eq4 | 177600em4 | \([0, -1, 0, -120925633, -4684151976863]\) | \(-47744008200656797609/2286529541015625000\) | \(-9365625000000000000000000000\) | \([2]\) | \(106168320\) | \(4.0472\) |
Rank
sage: E.rank()
The elliptic curves in class 177600.eq have rank \(0\).
Complex multiplication
The elliptic curves in class 177600.eq do not have complex multiplication.Modular form 177600.2.a.eq
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.