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SageMath
sage: E = EllipticCurve("6270.l1")
sage: E.isogeny_class()
Elliptic curves in class 6270.l
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | Torsion structure | Modular degree | Optimality |
---|---|---|---|---|---|
6270.l1 | 6270l7 | [1, 0, 1, -21424283, 37908785126] | [2] | 774144 | |
6270.l2 | 6270l4 | [1, 0, 1, -21384008, 38059355306] | [6] | 258048 | |
6270.l3 | 6270l6 | [1, 0, 1, -2257883, -324349594] | [2, 2] | 387072 | |
6270.l4 | 6270l3 | [1, 0, 1, -1745883, -886935194] | [2] | 193536 | |
6270.l5 | 6270l2 | [1, 0, 1, -1336508, 594587306] | [2, 6] | 129024 | |
6270.l6 | 6270l5 | [1, 0, 1, -1289008, 638819306] | [6] | 258048 | |
6270.l7 | 6270l1 | [1, 0, 1, -86508, 8587306] | [6] | 64512 | \(\Gamma_0(N)\)-optimal |
6270.l8 | 6270l8 | [1, 0, 1, 8716517, -2549957914] | [2] | 774144 |
Rank
sage: E.rank()
The elliptic curves in class 6270.l have rank \(0\).
Modular form 6270.2.a.l
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 3 & 2 & 4 & 6 & 12 & 12 & 4 \\ 3 & 1 & 6 & 12 & 2 & 4 & 4 & 12 \\ 2 & 6 & 1 & 2 & 3 & 6 & 6 & 2 \\ 4 & 12 & 2 & 1 & 6 & 12 & 3 & 4 \\ 6 & 2 & 3 & 6 & 1 & 2 & 2 & 6 \\ 12 & 4 & 6 & 12 & 2 & 1 & 4 & 3 \\ 12 & 4 & 6 & 3 & 2 & 4 & 1 & 12 \\ 4 & 12 & 2 & 4 & 6 & 3 & 12 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.