Properties

Label 106470ce
Number of curves 8
Conductor 106470
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("106470.cg1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 106470ce

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
106470.cg7 106470ce1 [1, -1, 0, -756729, -253085715] [2] 1769472 \(\Gamma_0(N)\)-optimal
106470.cg6 106470ce2 [1, -1, 0, -878409, -166133187] [2, 2] 3538944  
106470.cg5 106470ce3 [1, -1, 0, -2239704, 981046080] [2] 5308416  
106470.cg8 106470ce4 [1, -1, 0, 2924091, -1222467687] [2] 7077888  
106470.cg4 106470ce5 [1, -1, 0, -6627789, 6453702945] [2] 7077888  
106470.cg2 106470ce6 [1, -1, 0, -33389784, 74264724288] [2, 2] 10616832  
106470.cg3 106470ce7 [1, -1, 0, -30956184, 85547380608] [2] 21233664  
106470.cg1 106470ce8 [1, -1, 0, -534224664, 4752763672320] [2] 21233664  

Rank

sage: E.rank()
 

The elliptic curves in class 106470ce have rank \(1\).

Modular form 106470.2.a.cg

sage: E.q_eigenform(10)
 
\( q - q^{2} + q^{4} + q^{5} - q^{7} - q^{8} - q^{10} + q^{14} + q^{16} + 6q^{17} - 8q^{19} + O(q^{20}) \)

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.