Properties

Label 448f
Number of curves 66
Conductor 448448
CM no
Rank 00
Graph

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Show commands: SageMath
E = EllipticCurve("f1")
 
E.isogeny_class()
 

Elliptic curves in class 448f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
448.a5 448f1 [0,1,0,33,161][0, 1, 0, -33, -161] 15625/28-15625/28 7340032-7340032 [2][2] 6464 0.00763590.0076359 Γ0(N)\Gamma_0(N)-optimal
448.a4 448f2 [0,1,0,673,6945][0, 1, 0, -673, -6945] 128787625/98128787625/98 2569011225690112 [2][2] 128128 0.354210.35421  
448.a6 448f3 [0,1,0,287,3231][0, 1, 0, 287, 3231] 9938375/219529938375/21952 5754585088-5754585088 [2][2] 192192 0.556940.55694  
448.a3 448f4 [0,1,0,2273,33439][0, 1, 0, -2273, 33439] 4956477625/9411924956477625/941192 246727835648246727835648 [2][2] 384384 0.903520.90352  
448.a2 448f5 [0,1,0,10913,436447][0, 1, 0, -10913, 436447] 548347731625/1835008-548347731625/1835008 481036337152-481036337152 [2][2] 576576 1.10621.1062  
448.a1 448f6 [0,1,0,174753,28059871][0, 1, 0, -174753, 28059871] 2251439055699625/250882251439055699625/25088 65766686726576668672 [2][2] 11521152 1.45281.4528  

Rank

sage: E.rank()
 

The elliptic curves in class 448f have rank 00.

Complex multiplication

The elliptic curves in class 448f do not have complex multiplication.

Modular form 448.2.a.f

sage: E.q_eigenform(10)
 
q2q3q7+q9+4q13+6q17+2q19+O(q20)q - 2 q^{3} - q^{7} + q^{9} + 4 q^{13} + 6 q^{17} + 2 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.

(1236918216318936123663216391836121896321)\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 6 & 9 & 18 \\ 2 & 1 & 6 & 3 & 18 & 9 \\ 3 & 6 & 1 & 2 & 3 & 6 \\ 6 & 3 & 2 & 1 & 6 & 3 \\ 9 & 18 & 3 & 6 & 1 & 2 \\ 18 & 9 & 6 & 3 & 2 & 1 \end{array}\right)

Isogeny graph

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

The vertices are labelled with Cremona labels.