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Properties

Label 301665.bw
Number of curves $4$
Conductor $301665$
CM no
Rank $0$
Graph

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

Elliptic curves in class 301665.bw

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
301665.bw1 301665bw3 \([1, 0, 1, -1970978559, 33679748633221]\) \(175432195151291504528610481/1783697187806625\) \(8609565639379707809625\) \([2]\) \(117669888\) \(3.7837\)  
301665.bw2 301665bw4 \([1, 0, 1, -176832309, 24339794221]\) \(126691782970490027070481/73263917649445443375\) \(353630937085602111091440375\) \([2]\) \(117669888\) \(3.7837\)  
301665.bw3 301665bw2 \([1, 0, 1, -123280434, 525392557471]\) \(42928533940472967840481/136559534445140625\) \(659146789895614775015625\) \([2, 2]\) \(58834944\) \(3.4371\)  
301665.bw4 301665bw1 \([1, 0, 1, -4452309, 15192119971]\) \(-2022177859966590481/19487441162109375\) \(-94062156388239990234375\) \([2]\) \(29417472\) \(3.0905\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 301665.bw have rank \(0\).

Complex multiplication

The elliptic curves in class 301665.bw do not have complex multiplication.

Modular form 301665.2.a.bw

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{3} - q^{4} - q^{5} + q^{6} + q^{7} - 3 q^{8} + q^{9} - q^{10} - 4 q^{11} - q^{12} + q^{14} - q^{15} - q^{16} + q^{17} + q^{18} + O(q^{20})\) 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.