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SageMath
E = EllipticCurve("by1")
E.isogeny_class()
Elliptic curves in class 372810.by
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
372810.by1 | 372810by3 | \([1, 1, 1, -9914440, -5223806203]\) | \(4465136636671380769/2096375976562500\) | \(50601419784219726562500\) | \([2]\) | \(39813120\) | \(3.0508\) | |
372810.by2 | 372810by1 | \([1, 1, 1, -5076580, 4400226725]\) | \(599437478278595809/33854760000\) | \(817171605478440000\) | \([2]\) | \(13271040\) | \(2.5015\) | \(\Gamma_0(N)\)-optimal |
372810.by3 | 372810by2 | \([1, 1, 1, -4787580, 4923663525]\) | \(-502780379797811809/143268096832200\) | \(-3458143572785908921800\) | \([2]\) | \(26542080\) | \(2.8481\) | |
372810.by4 | 372810by4 | \([1, 1, 1, 35241810, -39506431203]\) | \(200541749524551119231/144008551960031250\) | \(-3476016359525339539031250\) | \([2]\) | \(79626240\) | \(3.3974\) |
Rank
sage: E.rank()
The elliptic curves in class 372810.by have rank \(0\).
Complex multiplication
The elliptic curves in class 372810.by do not have complex multiplication.Modular form 372810.2.a.by
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.
\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with LMFDB labels.