Properties

Label 41070b
Number of curves $6$
Conductor $41070$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("41070.d1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 41070b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
41070.d5 41070b1 [1, 1, 0, -1156833, 1648134837] [2] 2363904 \(\Gamma_0(N)\)-optimal
41070.d4 41070b2 [1, 1, 0, -29193953, 60576553653] [2, 2] 4727808  
41070.d3 41070b3 [1, 1, 0, -40145953, 10979326453] [2, 2] 9455616  
41070.d1 41070b4 [1, 1, 0, -466835873, 3882153327477] [2] 9455616  
41070.d6 41070b5 [1, 1, 0, 159454247, 87745563373] [2] 18911232  
41070.d2 41070b6 [1, 1, 0, -414978153, -3239790411267] [2] 18911232  

Rank

sage: E.rank()
 

The elliptic curves in class 41070b have rank \(1\).

Modular form 41070.2.a.d

sage: E.q_eigenform(10)
 
\( q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} - q^{8} + q^{9} + q^{10} + 4q^{11} - q^{12} + 2q^{13} + q^{15} + q^{16} - 2q^{17} - q^{18} - 4q^{19} + O(q^{20}) \)

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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.