Properties

Label 250470dd
Number of curves $6$
Conductor $250470$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("250470.dd1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 250470dd

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
250470.dd5 250470dd1 [1, -1, 1, -457403, 130745387] [2] 3932160 \(\Gamma_0(N)\)-optimal
250470.dd4 250470dd2 [1, -1, 1, -7514123, 7929832331] [2, 2] 7864320  
250470.dd1 250470dd3 [1, -1, 1, -120225623, 507422115731] [2] 15728640  
250470.dd3 250470dd4 [1, -1, 1, -7710143, 7494432707] [2, 2] 15728640  
250470.dd6 250470dd5 [1, -1, 1, 9572287, 36300787031] [2] 31457280  
250470.dd2 250470dd6 [1, -1, 1, -28128893, -49179849793] [2] 31457280  

Rank

sage: E.rank()
 

The elliptic curves in class 250470dd have rank \(1\).

Modular form 250470.2.a.dd

sage: E.q_eigenform(10)
 
\( q + q^{2} + q^{4} - q^{5} + q^{8} - q^{10} + 2q^{13} + q^{16} - 6q^{17} - 4q^{19} + O(q^{20}) \)

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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.