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SageMath
sage: E = EllipticCurve("dz1")
sage: E.isogeny_class()
Elliptic curves in class 47040.dz
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | Torsion structure | Modular degree | Optimality |
---|---|---|---|---|---|
47040.dz1 | 47040gk4 | [0, 1, 0, -1171361, 487569375] | [2] | 589824 | |
47040.dz2 | 47040gk2 | [0, 1, 0, -73761, 7479135] | [2, 2] | 294912 | |
47040.dz3 | 47040gk1 | [0, 1, 0, -11041, -285601] | [2] | 147456 | \(\Gamma_0(N)\)-optimal |
47040.dz4 | 47040gk3 | [0, 1, 0, 20319, 25335519] | [2] | 589824 |
Rank
sage: E.rank()
The elliptic curves in class 47040.dz have rank \(0\).
Complex multiplication
The elliptic curves in class 47040.dz do not have complex multiplication.Modular form 47040.2.a.dz
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.