Properties

Label 372810.by
Number of curves $4$
Conductor $372810$
CM no
Rank $0$
Graph

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Show commands: 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)
 
\(q + q^{2} - q^{3} + q^{4} + q^{5} - q^{6} - 2 q^{7} + q^{8} + q^{9} + q^{10} - q^{12} + 2 q^{13} - 2 q^{14} - q^{15} + q^{16} + q^{18} + 8 q^{19} + O(q^{20})\) Copy content Toggle raw display

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.