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SageMath
E = EllipticCurve("cy1")
E.isogeny_class()
Elliptic curves in class 206910.cy
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
206910.cy1 | 206910be3 | \([1, -1, 1, -3310583, 2319315221]\) | \(3107086841064961/570\) | \(736136742330\) | \([2]\) | \(3932160\) | \(2.1128\) | |
206910.cy2 | 206910be4 | \([1, -1, 1, -239603, 24077837]\) | \(1177918188481/488703750\) | \(631145239455183750\) | \([2]\) | \(3932160\) | \(2.1128\) | |
206910.cy3 | 206910be2 | \([1, -1, 1, -206933, 36270281]\) | \(758800078561/324900\) | \(419597943128100\) | \([2, 2]\) | \(1966080\) | \(1.7662\) | |
206910.cy4 | 206910be1 | \([1, -1, 1, -10913, 751457]\) | \(-111284641/123120\) | \(-159005536343280\) | \([2]\) | \(983040\) | \(1.4196\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 206910.cy have rank \(1\).
Complex multiplication
The elliptic curves in class 206910.cy do not have complex multiplication.Modular form 206910.2.a.cy
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.
\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.