Properties

Label 235200qq
Number of curves 44
Conductor 235200235200
CM no
Rank 11
Graph

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Show commands: SageMath
E = EllipticCurve("qq1")
 
E.isogeny_class()
 

Elliptic curves in class 235200qq

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
235200.qq4 235200qq1 [0,1,0,1416508,977960738][0, 1, 0, -1416508, 977960738] 2671731885376/1969120125-2671731885376/1969120125 231665013586125000000-231665013586125000000 [2][2] 70778887077888 2.60712.6071 Γ0(N)\Gamma_0(N)-optimal
235200.qq3 235200qq2 [0,1,0,25726633,50205963863][0, 1, 0, -25726633, 50205963863] 250094631024064/62015625250094631024064/62015625 466948881000000000000466948881000000000000 [2,2][2, 2] 1415577614155776 2.95362.9536  
235200.qq1 235200qq3 [0,1,0,411601633,3213995088863][0, 1, 0, -411601633, 3213995088863] 128025588102048008/7875128025588102048008/7875 474360768000000000474360768000000000 [2][2] 2831155228311552 3.30023.3002  
235200.qq2 235200qq4 [0,1,0,28813633,37398000863][0, 1, 0, -28813633, 37398000863] 43919722445768/1538085937543919722445768/15380859375 926485875000000000000000926485875000000000000000 [2][2] 2831155228311552 3.30023.3002  

Rank

sage: E.rank()
 

The elliptic curves in class 235200qq have rank 11.

Complex multiplication

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

Modular form 235200.2.a.qq

sage: E.q_eigenform(10)
 
q+q3+q94q11+2q132q17+8q19+O(q20)q + q^{3} + q^{9} - 4 q^{11} + 2 q^{13} - 2 q^{17} + 8 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.

(1244212242144241)\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)

Isogeny graph

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

The vertices are labelled with Cremona labels.