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SageMath
sage: E = EllipticCurve("n1")
sage: E.isogeny_class()
Elliptic curves in class 1800.n
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | Torsion structure | Modular degree | Optimality |
---|---|---|---|---|---|
1800.n1 | 1800r5 | [0, 0, 0, -720075, 235187750] | [2] | 12288 | |
1800.n2 | 1800r3 | [0, 0, 0, -45075, 3662750] | [2, 2] | 6144 | |
1800.n3 | 1800r6 | [0, 0, 0, -18075, 8009750] | [2] | 12288 | |
1800.n4 | 1800r2 | [0, 0, 0, -4575, -22750] | [2, 2] | 3072 | |
1800.n5 | 1800r1 | [0, 0, 0, -3450, -77875] | [2] | 1536 | \(\Gamma_0(N)\)-optimal |
1800.n6 | 1800r4 | [0, 0, 0, 17925, -180250] | [2] | 6144 |
Rank
sage: E.rank()
The elliptic curves in class 1800.n have rank \(1\).
Complex multiplication
The elliptic curves in class 1800.n do not have complex multiplication.Modular form 1800.2.a.n
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.
\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.