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SageMath
E = EllipticCurve("ey1")
E.isogeny_class()
Elliptic curves in class 400752.ey
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
400752.ey1 | 400752ey5 | \([0, 0, 0, -1622244459, 25149106658522]\) | \(89254274298475942657/17457\) | \(92344960347475968\) | \([4]\) | \(62914560\) | \(3.5565\) | \(\Gamma_0(N)\)-optimal* |
400752.ey2 | 400752ey3 | \([0, 0, 0, -101390619, 392952021770]\) | \(21790813729717297/304746849\) | \(1612065972785887973376\) | \([2, 2]\) | \(31457280\) | \(3.2099\) | \(\Gamma_0(N)\)-optimal* |
400752.ey3 | 400752ey6 | \([0, 0, 0, -98515659, 416284622138]\) | \(-19989223566735457/2584262514273\) | \(-13670368299708353829015552\) | \([2]\) | \(62914560\) | \(3.5565\) | |
400752.ey4 | 400752ey4 | \([0, 0, 0, -24550779, -40606896022]\) | \(309368403125137/44372288367\) | \(234722873906019600887808\) | \([2]\) | \(31457280\) | \(3.2099\) | |
400752.ey5 | 400752ey2 | \([0, 0, 0, -6516939, 5772533690]\) | \(5786435182177/627352209\) | \(3318600840007243862016\) | \([2, 2]\) | \(15728640\) | \(2.8634\) | \(\Gamma_0(N)\)-optimal* |
400752.ey6 | 400752ey1 | \([0, 0, 0, 539781, 447532778]\) | \(3288008303/18259263\) | \(-96588813525183885312\) | \([2]\) | \(7864320\) | \(2.5168\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 400752.ey have rank \(0\).
Complex multiplication
The elliptic curves in class 400752.ey do not have complex multiplication.Modular form 400752.2.a.ey
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 8 & 4 & 8 \\ 8 & 4 & 8 & 1 & 2 & 4 \\ 4 & 2 & 4 & 2 & 1 & 2 \\ 8 & 4 & 8 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.