Properties

Label 444675bw
Number of curves $4$
Conductor $444675$
CM no
Rank $1$
Graph

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

Elliptic curves in class 444675bw

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
444675.bw3 444675bw1 \([1, 1, 1, -1633563, -641557344]\) \(148035889/31185\) \(101557061298054140625\) \([2]\) \(13271040\) \(2.5536\) \(\Gamma_0(N)\)-optimal*
444675.bw2 444675bw2 \([1, 1, 1, -8303688, 8643256656]\) \(19443408769/1334025\) \(4344385399972316015625\) \([2, 2]\) \(26542080\) \(2.9002\) \(\Gamma_0(N)\)-optimal*
444675.bw1 444675bw3 \([1, 1, 1, -130589313, 574336557906]\) \(75627935783569/396165\) \(1290150815749354453125\) \([2]\) \(53084160\) \(3.2468\) \(\Gamma_0(N)\)-optimal*
444675.bw4 444675bw4 \([1, 1, 1, 7259937, 37311453906]\) \(12994449551/192163125\) \(-625798373091250283203125\) \([2]\) \(53084160\) \(3.2468\)  
*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 444675bw1.

Rank

sage: E.rank()
 

The elliptic curves in class 444675bw have rank \(1\).

Complex multiplication

The elliptic curves in class 444675bw do not have complex multiplication.

Modular form 444675.2.a.bw

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{3} - q^{4} + q^{6} + 3 q^{8} + q^{9} + q^{12} + 2 q^{13} - q^{16} - 6 q^{17} - q^{18} + 4 q^{19} + O(q^{20})\) 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 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.