Properties

Label 221952.bi
Number of curves 44
Conductor 221952221952
CM no
Rank 00
Graph

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Copy content sage:E = EllipticCurve("bi1") E.isogeny_class()
 

Elliptic curves in class 221952.bi

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
221952.bi1 221952ba4 [0,1,0,748317,249407685][0, 1, 0, -748317, -249407685] 58591911104/24358591911104/243 192198386221056192198386221056 [2][2] 16384001638400 1.95021.9502  
221952.bi2 221952ba3 [0,1,0,46047,4034547][0, 1, 0, -46047, -4034547] 873722816/59049-873722816/59049 729753247683072-729753247683072 [2][2] 819200819200 1.60361.6036  
221952.bi3 221952ba2 [0,1,0,8477,288315][0, 1, 0, -8477, 288315] 85184/385184/3 23728195829762372819582976 [2][2] 327680327680 1.14551.1455  
221952.bi4 221952ba1 [0,1,0,193,16077][0, 1, 0, 193, 16077] 64/964/9 111225917952-111225917952 [2][2] 163840163840 0.798890.79889 Γ0(N)\Gamma_0(N)-optimal

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 221952.bi have rank 00.

L-function data

 
Bad L-factors:
Prime L-Factor
2211
331T1 - T
171711
 
Good L-factors:
Prime L-Factor Isogeny Class over Fp\mathbb{F}_p
55 12T+5T2 1 - 2 T + 5 T^{2} 1.5.ac
77 1+2T+7T2 1 + 2 T + 7 T^{2} 1.7.c
1111 1+11T2 1 + 11 T^{2} 1.11.a
1313 1+4T+13T2 1 + 4 T + 13 T^{2} 1.13.e
1919 14T+19T2 1 - 4 T + 19 T^{2} 1.19.ae
2323 14T+23T2 1 - 4 T + 23 T^{2} 1.23.ae
2929 16T+29T2 1 - 6 T + 29 T^{2} 1.29.ag
\cdots\cdots\cdots
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 221952.bi do not have complex multiplication.

Modular form 221952.2.a.bi

Copy content sage:E.q_eigenform(10)
 
q+q3+2q52q7+q94q13+2q15+4q19+O(q20)q + q^{3} + 2 q^{5} - 2 q^{7} + q^{9} - 4 q^{13} + 2 q^{15} + 4 q^{19} + O(q^{20}) Copy content Toggle raw display

Isogeny matrix

Copy content 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 LMFDB numbering.

(12510211055101210521)\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)

Isogeny graph

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

The vertices are labelled with LMFDB labels.