Properties

Label 338130y
Number of curves 8
Conductor 338130
CM no
Rank 1
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("338130.y1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 338130y

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
338130.y7 338130y1 [1, -1, 0, -5452075800, -149202592145600] [2] 637009920 \(\Gamma_0(N)\)-optimal
338130.y6 338130y2 [1, -1, 0, -14454448920, 470482963468096] [2, 2] 1274019840  
338130.y5 338130y3 [1, -1, 0, -67087036440, 6644606911560256] [2] 1911029760  
338130.y4 338130y4 [1, -1, 0, -211651864920, 37473670967754496] [2] 2548039680  
338130.y8 338130y5 [1, -1, 0, 38704997160, 3127062489980800] [2] 2548039680  
338130.y2 338130y6 [1, -1, 0, -1071449453160, 426880086563858800] [2, 2] 3822059520  
338130.y1 338130y7 [1, -1, 0, -17143191015660, 27320313468836296300L] [2] 7644119040  
338130.y3 338130y8 [1, -1, 0, -1069506558180, 428505349689527476] [2] 7644119040  

Rank

sage: E.rank()
 

The elliptic curves in class 338130y have rank \(1\).

Modular form 338130.2.a.y

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

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.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.