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SageMath
E = EllipticCurve("m1")
E.isogeny_class()
Elliptic curves in class 33810m
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
33810.c3 | 33810m1 | \([1, 1, 0, -10789923, -13646414067]\) | \(1180838681727016392361/692428800000\) | \(81463555891200000\) | \([2]\) | \(1474560\) | \(2.5690\) | \(\Gamma_0(N)\)-optimal |
33810.c2 | 33810m2 | \([1, 1, 0, -10852643, -13479817203]\) | \(1201550658189465626281/28577902500000000\) | \(3362161651222500000000\) | \([2, 2]\) | \(2949120\) | \(2.9156\) | |
33810.c4 | 33810m3 | \([1, 1, 0, 1397357, -42176667203]\) | \(2564821295690373719/6533572090396050000\) | \(-768668222863004886450000\) | \([2]\) | \(5898240\) | \(3.2622\) | |
33810.c1 | 33810m4 | \([1, 1, 0, -24106163, 25880486493]\) | \(13167998447866683762601/5158996582031250000\) | \(606950788879394531250000\) | \([2]\) | \(5898240\) | \(3.2622\) |
Rank
sage: E.rank()
The elliptic curves in class 33810m have rank \(1\).
Complex multiplication
The elliptic curves in class 33810m do not have complex multiplication.Modular form 33810.2.a.m
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 Cremona 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 Cremona labels.