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SageMath
sage: E = EllipticCurve("210.e1")
sage: E.isogeny_class()
Elliptic curves in class 210e
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | Torsion structure | Modular degree | Optimality |
---|---|---|---|---|---|
210.e7 | 210e1 | [1, 0, 0, 210, 900] | [8] | 128 | \(\Gamma_0(N)\)-optimal |
210.e6 | 210e2 | [1, 0, 0, -1070, 7812] | [2, 8] | 256 | |
210.e5 | 210e3 | [1, 0, 0, -7550, -247500] | [2, 4] | 512 | |
210.e4 | 210e4 | [1, 0, 0, -15070, 710612] | [8] | 512 | |
210.e2 | 210e5 | [1, 0, 0, -120050, -16020000] | [2, 2] | 1024 | |
210.e8 | 210e6 | [1, 0, 0, 1270, -789048] | [4] | 1024 | |
210.e1 | 210e7 | [1, 0, 0, -1920800, -1024800150] | [2] | 2048 | |
210.e3 | 210e8 | [1, 0, 0, -119300, -16229850] | [2] | 2048 |
Rank
sage: E.rank()
The elliptic curves in class 210e have rank \(0\).
Modular form 210.2.a.e
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 2 & 2 & 4 & 4 \\ 4 & 2 & 4 & 1 & 8 & 8 & 16 & 16 \\ 8 & 4 & 2 & 8 & 1 & 4 & 2 & 2 \\ 8 & 4 & 2 & 8 & 4 & 1 & 8 & 8 \\ 16 & 8 & 4 & 16 & 2 & 8 & 1 & 4 \\ 16 & 8 & 4 & 16 & 2 & 8 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.