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SageMath
E = EllipticCurve("cx1")
E.isogeny_class()
Elliptic curves in class 33810cx
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 33810.cz4 | 33810cx1 | \([1, 0, 0, 14130619, -37157688639]\) | \(2652277923951208297919/6605028468326400000\) | \(-777074994270132633600000\) | \([2]\) | \(7372800\) | \(3.2680\) | \(\Gamma_0(N)\)-optimal |
| 33810.cz3 | 33810cx2 | \([1, 0, 0, -118584901, -415104946495]\) | \(1567558142704512417614401/274462175610000000000\) | \(32290200498340890000000000\) | \([2, 2]\) | \(14745600\) | \(3.6146\) | |
| 33810.cz2 | 33810cx3 | \([1, 0, 0, -551533221, 4597484112801]\) | \(157706830105239346386477121/13650704956054687500000\) | \(1605991787374877929687500000\) | \([2]\) | \(29491200\) | \(3.9611\) | |
| 33810.cz1 | 33810cx4 | \([1, 0, 0, -1809084901, -29615787646495]\) | \(5565604209893236690185614401/229307220930246900000\) | \(26977765235222617538100000\) | \([2]\) | \(29491200\) | \(3.9611\) |
Rank
sage: E.rank()
The elliptic curves in class 33810cx have rank \(1\).
Complex multiplication
The elliptic curves in class 33810cx do not have complex multiplication.Modular form 33810.2.a.cx
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.
