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SageMath
E = EllipticCurve("w1")
E.isogeny_class()
Elliptic curves in class 3648.w
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 3648.w1 | 3648bg3 | \([0, 1, 0, -5089, -137569]\) | \(111223479026/3518667\) | \(461198721024\) | \([2]\) | \(3072\) | \(1.0128\) | |
| 3648.w2 | 3648bg2 | \([0, 1, 0, -769, 4991]\) | \(768400132/263169\) | \(17247043584\) | \([2, 2]\) | \(1536\) | \(0.66620\) | |
| 3648.w3 | 3648bg1 | \([0, 1, 0, -689, 6735]\) | \(2211014608/513\) | \(8404992\) | \([2]\) | \(768\) | \(0.31963\) | \(\Gamma_0(N)\)-optimal |
| 3648.w4 | 3648bg4 | \([0, 1, 0, 2271, 37215]\) | \(9878111854/10097379\) | \(-1323483660288\) | \([4]\) | \(3072\) | \(1.0128\) |
Rank
sage: E.rank()
The elliptic curves in class 3648.w have rank \(1\).
Complex multiplication
The elliptic curves in class 3648.w do not have complex multiplication.Modular form 3648.2.a.w
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.
