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SageMath
E = EllipticCurve("cr1")
E.isogeny_class()
Elliptic curves in class 361998.cr
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
361998.cr1 | 361998cr4 | \([1, -1, 1, -7725191, -8262476985]\) | \(14489843500598257/6246072\) | \(21978326880756792\) | \([2]\) | \(11796480\) | \(2.4783\) | |
361998.cr2 | 361998cr3 | \([1, -1, 1, -1032791, 213800487]\) | \(34623662831857/14438442312\) | \(50805178803908415432\) | \([2]\) | \(11796480\) | \(2.4783\) | |
361998.cr3 | 361998cr2 | \([1, -1, 1, -485231, -127657929]\) | \(3590714269297/73410624\) | \(258313175191116864\) | \([2, 2]\) | \(5898240\) | \(2.1318\) | |
361998.cr4 | 361998cr1 | \([1, -1, 1, 1489, -5977929]\) | \(103823/4386816\) | \(-15436081430654976\) | \([2]\) | \(2949120\) | \(1.7852\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 361998.cr have rank \(0\).
Complex multiplication
The elliptic curves in class 361998.cr do not have complex multiplication.Modular form 361998.2.a.cr
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.