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SageMath
sage: E = EllipticCurve("5415.j1")
sage: E.isogeny_class()
Elliptic curves in class 5415.j
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | Torsion structure | Modular degree | Optimality |
---|---|---|---|---|---|
5415.j1 | 5415k7 | [1, 0, 1, -779768, 264965501] | [2] | 27648 | |
5415.j2 | 5415k5 | [1, 0, 1, -48743, 4135781] | [2, 2] | 13824 | |
5415.j3 | 5415k8 | [1, 0, 1, -39718, 5716961] | [2] | 27648 | |
5415.j4 | 5415k3 | [1, 0, 1, -28888, -1892197] | [2] | 6912 | |
5415.j5 | 5415k4 | [1, 0, 1, -3618, 38431] | [2, 2] | 6912 | |
5415.j6 | 5415k2 | [1, 0, 1, -1813, -29437] | [2, 2] | 3456 | |
5415.j7 | 5415k1 | [1, 0, 1, -8, -1279] | [2] | 1728 | \(\Gamma_0(N)\)-optimal |
5415.j8 | 5415k6 | [1, 0, 1, 12627, 291853] | [2] | 13824 |
Rank
sage: E.rank()
The elliptic curves in class 5415.j have rank \(0\).
Modular form 5415.2.a.j
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.
\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 16 & 4 & 8 & 16 & 8 \\ 2 & 1 & 2 & 8 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 16 & 4 & 8 & 16 & 8 \\ 16 & 8 & 16 & 1 & 4 & 2 & 4 & 8 \\ 4 & 2 & 4 & 4 & 1 & 2 & 4 & 2 \\ 8 & 4 & 8 & 2 & 2 & 1 & 2 & 4 \\ 16 & 8 & 16 & 4 & 4 & 2 & 1 & 8 \\ 8 & 4 & 8 & 8 & 2 & 4 & 8 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.