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SageMath
E = EllipticCurve("n1")
E.isogeny_class()
Elliptic curves in class 1800.n
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 1800.n1 | 1800r5 | \([0, 0, 0, -720075, 235187750]\) | \(1770025017602/75\) | \(1749600000000\) | \([2]\) | \(12288\) | \(1.8340\) | |
| 1800.n2 | 1800r3 | \([0, 0, 0, -45075, 3662750]\) | \(868327204/5625\) | \(65610000000000\) | \([2, 2]\) | \(6144\) | \(1.4874\) | |
| 1800.n3 | 1800r6 | \([0, 0, 0, -18075, 8009750]\) | \(-27995042/1171875\) | \(-27337500000000000\) | \([2]\) | \(12288\) | \(1.8340\) | |
| 1800.n4 | 1800r2 | \([0, 0, 0, -4575, -22750]\) | \(3631696/2025\) | \(5904900000000\) | \([2, 2]\) | \(3072\) | \(1.1409\) | |
| 1800.n5 | 1800r1 | \([0, 0, 0, -3450, -77875]\) | \(24918016/45\) | \(8201250000\) | \([2]\) | \(1536\) | \(0.79430\) | \(\Gamma_0(N)\)-optimal |
| 1800.n6 | 1800r4 | \([0, 0, 0, 17925, -180250]\) | \(54607676/32805\) | \(-382637520000000\) | \([2]\) | \(6144\) | \(1.4874\) |
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.
