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SageMath
E = EllipticCurve("j1")
E.isogeny_class()
Elliptic curves in class 650j
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
650.l2 | 650j1 | \([1, 1, 1, -813, 8531]\) | \(3803721481/26000\) | \(406250000\) | \([2]\) | \(576\) | \(0.48586\) | \(\Gamma_0(N)\)-optimal |
650.l3 | 650j2 | \([1, 1, 1, -313, 19531]\) | \(-217081801/10562500\) | \(-165039062500\) | \([2]\) | \(1152\) | \(0.83244\) | |
650.l1 | 650j3 | \([1, 1, 1, -5188, -140219]\) | \(988345570681/44994560\) | \(703040000000\) | \([2]\) | \(1728\) | \(1.0352\) | |
650.l4 | 650j4 | \([1, 1, 1, 2812, -524219]\) | \(157376536199/7722894400\) | \(-120670225000000\) | \([2]\) | \(3456\) | \(1.3817\) |
Rank
sage: E.rank()
The elliptic curves in class 650j have rank \(0\).
Complex multiplication
The elliptic curves in class 650j do not have complex multiplication.Modular form 650.2.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 Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.