Properties

Label 74970bx
Number of curves $8$
Conductor $74970$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("bx1")
 
E.isogeny_class()
 

Elliptic curves in class 74970bx

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
74970.bf6 74970bx1 \([1, -1, 0, -35289, 3143853]\) \(-56667352321/16711680\) \(-1433295968993280\) \([2]\) \(393216\) \(1.6227\) \(\Gamma_0(N)\)-optimal
74970.bf5 74970bx2 \([1, -1, 0, -599769, 178922925]\) \(278202094583041/16646400\) \(1427697156614400\) \([2, 2]\) \(786432\) \(1.9693\)  
74970.bf4 74970bx3 \([1, -1, 0, -635049, 156717693]\) \(330240275458561/67652010000\) \(5802250475553210000\) \([2, 2]\) \(1572864\) \(2.3159\)  
74970.bf2 74970bx4 \([1, -1, 0, -9596169, 11444215005]\) \(1139466686381936641/4080\) \(349925773680\) \([2]\) \(1572864\) \(2.3159\)  
74970.bf7 74970bx5 \([1, -1, 0, 1349451, 939007593]\) \(3168685387909439/6278181696900\) \(-538455291076310724900\) \([2]\) \(3145728\) \(2.6624\)  
74970.bf3 74970bx6 \([1, -1, 0, -3184029, -2047130415]\) \(41623544884956481/2962701562500\) \(254099420696264062500\) \([2, 2]\) \(3145728\) \(2.6624\)  
74970.bf8 74970bx7 \([1, -1, 0, 2888541, -8937068337]\) \(31077313442863199/420227050781250\) \(-36041244084777832031250\) \([2]\) \(6291456\) \(3.0090\)  
74970.bf1 74970bx8 \([1, -1, 0, -50040279, -136234059165]\) \(161572377633716256481/914742821250\) \(78453943491208871250\) \([2]\) \(6291456\) \(3.0090\)  

Rank

sage: E.rank()
 

The elliptic curves in class 74970bx have rank \(1\).

Complex multiplication

The elliptic curves in class 74970bx do not have complex multiplication.

Modular form 74970.2.a.bx

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{4} + q^{5} - q^{8} - q^{10} - 4 q^{11} + 2 q^{13} + q^{16} + q^{17} - 4 q^{19} + O(q^{20})\) 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 Cremona numbering.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 2 & 2 & 4 & 4 \\ 4 & 2 & 4 & 1 & 8 & 8 & 16 & 16 \\ 8 & 4 & 2 & 8 & 1 & 4 & 8 & 8 \\ 8 & 4 & 2 & 8 & 4 & 1 & 2 & 2 \\ 16 & 8 & 4 & 16 & 8 & 2 & 1 & 4 \\ 16 & 8 & 4 & 16 & 8 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.