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SageMath
sage: E = EllipticCurve("25886.d1")
sage: E.isogeny_class()
Elliptic curves in class 25886a
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | Torsion structure | Modular degree | Optimality |
---|---|---|---|---|---|
25886.d5 | 25886a1 | [1, 1, 1, -963, -23683] | [2] | 26208 | \(\Gamma_0(N)\)-optimal |
25886.d4 | 25886a2 | [1, 1, 1, -19453, -1051727] | [2] | 52416 | |
25886.d6 | 25886a3 | [1, 1, 1, 8282, 490339] | [2] | 78624 | |
25886.d3 | 25886a4 | [1, 1, 1, -65678, 5282947] | [2] | 157248 | |
25886.d2 | 25886a5 | [1, 1, 1, -315293, 68208115] | [2] | 235872 | |
25886.d1 | 25886a6 | [1, 1, 1, -5048733, 4364278259] | [2] | 471744 |
Rank
sage: E.rank()
The elliptic curves in class 25886a have rank \(1\).
Modular form 25886.2.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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 6 & 9 & 18 \\ 2 & 1 & 6 & 3 & 18 & 9 \\ 3 & 6 & 1 & 2 & 3 & 6 \\ 6 & 3 & 2 & 1 & 6 & 3 \\ 9 & 18 & 3 & 6 & 1 & 2 \\ 18 & 9 & 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.