Properties

Label 494190.bn
Number of curves $4$
Conductor $494190$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("bn1")
 
E.isogeny_class()
 

Elliptic curves in class 494190.bn

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
494190.bn1 494190bn4 \([1, -1, 0, -7907094, -8556046862]\) \(3107086841064961/570\) \(10029884046570\) \([2]\) \(15728640\) \(2.3304\)  
494190.bn2 494190bn3 \([1, -1, 0, -572274, -88512170]\) \(1177918188481/488703750\) \(8599371834427953750\) \([2]\) \(15728640\) \(2.3304\) \(\Gamma_0(N)\)-optimal*
494190.bn3 494190bn2 \([1, -1, 0, -494244, -133566692]\) \(758800078561/324900\) \(5717033906544900\) \([2, 2]\) \(7864320\) \(1.9839\) \(\Gamma_0(N)\)-optimal*
494190.bn4 494190bn1 \([1, -1, 0, -26064, -2757200]\) \(-111284641/123120\) \(-2166454954059120\) \([2]\) \(3932160\) \(1.6373\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 0 curves highlighted, and conditionally curve 494190.bn1.

Rank

sage: E.rank()
 

The elliptic curves in class 494190.bn have rank \(1\).

Complex multiplication

The elliptic curves in class 494190.bn do not have complex multiplication.

Modular form 494190.2.a.bn

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{4} + q^{5} - 4 q^{7} - q^{8} - q^{10} - 4 q^{11} - 2 q^{13} + 4 q^{14} + q^{16} - q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrr} 1 & 4 & 2 & 4 \\ 4 & 1 & 2 & 4 \\ 2 & 2 & 1 & 2 \\ 4 & 4 & 2 & 1 \end{array}\right)\)

Isogeny graph

sage: E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with LMFDB labels.