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SageMath
E = EllipticCurve("bw1")
E.isogeny_class()
Elliptic curves in class 321552bw
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 321552.bw3 | 321552bw1 | \([0, 0, 0, -22251, -1037990]\) | \(408023180713/80247321\) | \(239617216548864\) | \([2]\) | \(786432\) | \(1.4758\) | \(\Gamma_0(N)\)-optimal |
| 321552.bw2 | 321552bw2 | \([0, 0, 0, -109371, 12988330]\) | \(48455467135993/3635004681\) | \(10854065817391104\) | \([2, 2]\) | \(1572864\) | \(1.8223\) | |
| 321552.bw1 | 321552bw3 | \([0, 0, 0, -1717131, 866065786]\) | \(187519537050946633/1186707753\) | \(3543490363133952\) | \([2]\) | \(3145728\) | \(2.1689\) | |
| 321552.bw4 | 321552bw4 | \([0, 0, 0, 104469, 57595354]\) | \(42227808999767/504359959257\) | \(-1506010768582053888\) | \([4]\) | \(3145728\) | \(2.1689\) |
Rank
sage: E.rank()
The elliptic curves in class 321552bw have rank \(1\).
Complex multiplication
The elliptic curves in class 321552bw do not have complex multiplication.Modular form 321552.2.a.bw
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.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
