Properties

Label 1305.f
Number of curves 22
Conductor 13051305
CM no
Rank 00
Graph

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Copy content sage:E = EllipticCurve("f1") E.isogeny_class()
 

Elliptic curves in class 1305.f

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1305.f1 1305g1 [1,1,0,24,37][1, -1, 0, -24, -37] 2146689/1452146689/145 105705105705 [2][2] 128128 0.28720-0.28720 Γ0(N)\Gamma_0(N)-optimal
1305.f2 1305g2 [1,1,0,21,190][1, -1, 0, 21, -190] 1367631/210251367631/21025 15327225-15327225 [2][2] 256256 0.0593780.059378  

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 1305.f have rank 00.

L-function data

 
Bad L-factors:
Prime L-Factor
3311
551T1 - T
29291T1 - T
 
Good L-factors:
Prime L-Factor Isogeny Class over Fp\mathbb{F}_p
22 1T+2T2 1 - T + 2 T^{2} 1.2.ab
77 1+2T+7T2 1 + 2 T + 7 T^{2} 1.7.c
1111 16T+11T2 1 - 6 T + 11 T^{2} 1.11.ag
1313 12T+13T2 1 - 2 T + 13 T^{2} 1.13.ac
1717 12T+17T2 1 - 2 T + 17 T^{2} 1.17.ac
1919 1+2T+19T2 1 + 2 T + 19 T^{2} 1.19.c
2323 1+2T+23T2 1 + 2 T + 23 T^{2} 1.23.c
\cdots\cdots\cdots
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 1305.f do not have complex multiplication.

Modular form 1305.2.a.f

Copy content sage:E.q_eigenform(10)
 
q+q2q4+q52q73q8+q10+6q11+2q132q14q16+2q172q19+O(q20)q + q^{2} - q^{4} + q^{5} - 2 q^{7} - 3 q^{8} + q^{10} + 6 q^{11} + 2 q^{13} - 2 q^{14} - q^{16} + 2 q^{17} - 2 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.

(1221)\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)

Isogeny graph

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

The vertices are labelled with LMFDB labels.