Properties

Label 40656bk
Number of curves 66
Conductor 4065640656
CM no
Rank 11
Graph

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Show commands: SageMath
E = EllipticCurve("bk1")
 
E.isogeny_class()
 

Elliptic curves in class 40656bk

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
40656.l6 40656bk1 [0,1,0,1896,7632][0, -1, 0, 1896, -7632] 103823/63103823/63 457147772928-457147772928 [2][2] 4096040960 0.926580.92658 Γ0(N)\Gamma_0(N)-optimal
40656.l5 40656bk2 [0,1,0,7784,54096][0, -1, 0, -7784, -54096] 7189057/39697189057/3969 2880030969446428800309694464 [2,2][2, 2] 8192081920 1.27321.2732  
40656.l3 40656bk3 [0,1,0,75544,7968688][0, -1, 0, -75544, 7968688] 6570725617/459276570725617/45927 333260726464512333260726464512 [2][2] 163840163840 1.61971.6197  
40656.l2 40656bk4 [0,1,0,94904,11205456][0, -1, 0, -94904, -11205456] 13027640977/2160913027640977/21609 156801686114304156801686114304 [2,2][2, 2] 163840163840 1.61971.6197  
40656.l4 40656bk5 [0,1,0,65864,18221520][0, -1, 0, -65864, -18221520] 4354703137/17294403-4354703137/17294403 125493616120147968-125493616120147968 [2][2] 327680327680 1.96631.9663  
40656.l1 40656bk6 [0,1,0,1517864,719270352][0, -1, 0, -1517864, -719270352] 53297461115137/14753297461115137/147 10666781368321066678136832 [2][2] 327680327680 1.96631.9663  

Rank

sage: E.rank()
 

The elliptic curves in class 40656bk have rank 11.

Complex multiplication

The elliptic curves in class 40656bk do not have complex multiplication.

Modular form 40656.2.a.bk

sage: E.q_eigenform(10)
 
qq32q5q7+q9+2q13+2q15+6q17+4q19+O(q20)q - q^{3} - 2 q^{5} - q^{7} + q^{9} + 2 q^{13} + 2 q^{15} + 6 q^{17} + 4 q^{19} + O(q^{20}) Copy content Toggle raw display

Isogeny matrix

sage: E.isogeny_class().matrix()
 

The i,ji,j entry is the smallest degree of a cyclic isogeny between the ii-th and jj-th curve in the isogeny class, in the Cremona numbering.

(124488212244421488424122848214848241)\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.