Properties

Label 441090.bw
Number of curves $4$
Conductor $441090$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("bw1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 441090.bw

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
441090.bw1 441090bw3 \([1, -1, 0, -12704184, 15960215740]\) \(64443098670429961/6032611833300\) \(21227215250959147041300\) \([2]\) \(66060288\) \(3.0225\) \(\Gamma_0(N)\)-optimal*
441090.bw2 441090bw2 \([1, -1, 0, -2817684, -1540866560]\) \(703093388853961/115124490000\) \(405093580925806890000\) \([2, 2]\) \(33030144\) \(2.6759\) \(\Gamma_0(N)\)-optimal*
441090.bw3 441090bw1 \([1, -1, 0, -2696004, -1703114672]\) \(615882348586441/21715200\) \(76410224518867200\) \([2]\) \(16515072\) \(2.3294\) \(\Gamma_0(N)\)-optimal*
441090.bw4 441090bw4 \([1, -1, 0, 5121936, -8659529852]\) \(4223169036960119/11647532812500\) \(-40984683415027157812500\) \([2]\) \(66060288\) \(3.0225\)  
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 3 curves highlighted, and conditionally curve 441090.bw1.

Rank

sage: E.rank()
 

The elliptic curves in class 441090.bw have rank \(1\).

Complex multiplication

The elliptic curves in class 441090.bw do not have complex multiplication.

Modular form 441090.2.a.bw

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{4} + q^{5} + 4 q^{7} - q^{8} - q^{10} - 4 q^{11} - 4 q^{14} + q^{16} + 6 q^{17} + 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.