Properties

Label 338.b
Number of curves 22
Conductor 338338
CM no
Rank 00
Graph

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Copy content sage:E = EllipticCurve("b1") E.isogeny_class()
 

Elliptic curves in class 338.b

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
338.b1 338d2 [1,1,0,54421,4945517][1, 1, 0, -54421, 4945517] 1680914269/32768-1680914269/32768 347488235454464-347488235454464 [][] 15601560 1.58361.5836  
338.b2 338d1 [1,1,0,504,13112][1, 1, 0, 504, -13112] 1331/81331/8 84835994984-84835994984 [][] 312312 0.778900.77890 Γ0(N)\Gamma_0(N)-optimal

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 338.b have rank 00.

L-function data

 
Bad L-factors:
Prime L-Factor
221+T1 + T
131311
 
Good L-factors:
Prime L-Factor Isogeny Class over Fp\mathbb{F}_p
33 1+T+3T2 1 + T + 3 T^{2} 1.3.b
55 13T+5T2 1 - 3 T + 5 T^{2} 1.5.ad
77 13T+7T2 1 - 3 T + 7 T^{2} 1.7.ad
1111 1+11T2 1 + 11 T^{2} 1.11.a
1717 1+3T+17T2 1 + 3 T + 17 T^{2} 1.17.d
1919 16T+19T2 1 - 6 T + 19 T^{2} 1.19.ag
2323 16T+23T2 1 - 6 T + 23 T^{2} 1.23.ag
2929 1+29T2 1 + 29 T^{2} 1.29.a
\cdots\cdots\cdots
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 338.b do not have complex multiplication.

Modular form 338.2.a.b

Copy content sage:E.q_eigenform(10)
 
qq2q3+q4+3q5+q6+3q7q82q93q10q123q143q15+q163q17+2q18+6q19+O(q20)q - q^{2} - q^{3} + q^{4} + 3 q^{5} + q^{6} + 3 q^{7} - q^{8} - 2 q^{9} - 3 q^{10} - q^{12} - 3 q^{14} - 3 q^{15} + q^{16} - 3 q^{17} + 2 q^{18} + 6 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.

(1551)\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)

Isogeny graph

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

The vertices are labelled with LMFDB labels.