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SageMath
E = EllipticCurve("ca1")
E.isogeny_class()
Elliptic curves in class 210210.ca
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 210210.ca1 | 210210dk3 | \([1, 0, 1, -1869278, 983534906]\) | \(6139836723518159689/3799803150\) | \(447043040794350\) | \([2]\) | \(3538944\) | \(2.1321\) | |
| 210210.ca2 | 210210dk4 | \([1, 0, 1, -263058, -29888582]\) | \(17111482619973769/6627044531250\) | \(779665162057031250\) | \([2]\) | \(3538944\) | \(2.1321\) | |
| 210210.ca3 | 210210dk2 | \([1, 0, 1, -117528, 15167506]\) | \(1525998818291689/37268302500\) | \(4384578520822500\) | \([2, 2]\) | \(1769472\) | \(1.7855\) | |
| 210210.ca4 | 210210dk1 | \([1, 0, 1, 1052, 748178]\) | \(1095912791/2055596400\) | \(-241838860863600\) | \([2]\) | \(884736\) | \(1.4389\) | \(\Gamma_0(N)\)-optimal |
Rank
sage: E.rank()
The elliptic curves in class 210210.ca have rank \(2\).
Complex multiplication
The elliptic curves in class 210210.ca do not have complex multiplication.Modular form 210210.2.a.ca
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.
