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SageMath
sage: E = EllipticCurve("d1")
sage: E.isogeny_class()
Elliptic curves in class 210d
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | Torsion structure | Modular degree | Optimality |
---|---|---|---|---|---|
210.a3 | 210d1 | [1, 1, 0, -3, -3] | [2] | 16 | \(\Gamma_0(N)\)-optimal |
210.a2 | 210d2 | [1, 1, 0, -23, 33] | [2, 2] | 32 | |
210.a1 | 210d3 | [1, 1, 0, -373, 2623] | [2] | 64 | |
210.a4 | 210d4 | [1, 1, 0, 7, 147] | [2] | 64 |
Rank
sage: E.rank()
The elliptic curves in class 210d have rank \(1\).
Complex multiplication
The elliptic curves in class 210d do not have complex multiplication.Modular form 210.2.a.d
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.