Properties

Label 235200qr
Number of curves 88
Conductor 235200235200
CM no
Rank 00
Graph

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Show commands: SageMath
E = EllipticCurve("qr1")
 
E.isogeny_class()
 

Elliptic curves in class 235200qr

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
235200.qr7 235200qr1 [0,1,0,16462367,19378152863][0, 1, 0, 16462367, 19378152863] 1023887723039/9289728001023887723039/928972800 447662984999731200000000-447662984999731200000000 [2][2] 2831155228311552 3.22553.2255 Γ0(N)\Gamma_0(N)-optimal
235200.qr6 235200qr2 [0,1,0,83889633,173418472863][0, 1, 0, -83889633, 173418472863] 135487869158881/51438240000135487869158881/51438240000 2478758911082496000000000024787589110824960000000000 [2,2][2, 2] 5662310456623104 3.57213.5721  
235200.qr4 235200qr3 [0,1,0,1181489633,15626528872863][0, 1, 0, -1181489633, 15626528872863] 378499465220294881/120530818800378499465220294881/120530818800 5808263291290091520000000058082632912900915200000000 [2][2] 113246208113246208 3.91873.9187  
235200.qr5 235200qr4 [0,1,0,591921633,5419505815137][0, 1, 0, -591921633, -5419505815137] 47595748626367201/121550625000047595748626367201/1215506250000 585740676326400000000000000585740676326400000000000000 [2,2][2, 2] 113246208113246208 3.91873.9187  
235200.qr8 235200qr5 [0,1,0,99566367,17323471735137][0, 1, 0, 99566367, -17323471735137] 226523624554079/269165039062500226523624554079/269165039062500 129708022500000000000000000000-129708022500000000000000000000 [2][2] 226492416226492416 4.26534.2653  
235200.qr2 235200qr6 [0,1,0,9411921633,351454565815137][0, 1, 0, -9411921633, -351454565815137] 191342053882402567201/129708022500191342053882402567201/129708022500 6250503839376384000000000062505038393763840000000000 [2,2][2, 2] 226492416226492416 4.26534.2653  
235200.qr3 235200qr7 [0,1,0,9353121633,356062545415137][0, 1, 0, -9353121633, -356062545415137] 187778242790732059201/4984939585440150-187778242790732059201/4984939585440150 2402194052249387857305600000000-2402194052249387857305600000000 [2][2] 452984832452984832 4.61184.6118  
235200.qr1 235200qr8 [0,1,0,150590721633,22492949306215137][0, 1, 0, -150590721633, -22492949306215137] 783736670177727068275201/360150783736670177727068275201/360150 173552792985600000000173552792985600000000 [2][2] 452984832452984832 4.61184.6118  

Rank

sage: E.rank()
 

The elliptic curves in class 235200qr have rank 00.

Complex multiplication

The elliptic curves in class 235200qr do not have complex multiplication.

Modular form 235200.2.a.qr

sage: E.q_eigenform(10)
 
q+q3+q94q11+2q13+2q174q19+O(q20)q + q^{3} + q^{9} - 4 q^{11} + 2 q^{13} + 2 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.

(124488161621224488421488161642412244848214888482412216816482141681648241)\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 8 & 8 & 16 & 16 \\ 4 & 2 & 4 & 1 & 2 & 2 & 4 & 4 \\ 8 & 4 & 8 & 2 & 1 & 4 & 8 & 8 \\ 8 & 4 & 8 & 2 & 4 & 1 & 2 & 2 \\ 16 & 8 & 16 & 4 & 8 & 2 & 1 & 4 \\ 16 & 8 & 16 & 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.