Properties

Label 5040bi
Number of curves 88
Conductor 50405040
CM no
Rank 00
Graph

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Show commands: SageMath
E = EllipticCurve("bi1")
 
E.isogeny_class()
 

Elliptic curves in class 5040bi

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
5040.k7 5040bi1 [0,0,0,30237,1524962][0, 0, 0, 30237, 1524962] 1023887723039/9289728001023887723039/928972800 2773897917235200-2773897917235200 [2][2] 2457624576 1.65061.6506 Γ0(N)\Gamma_0(N)-optimal
5040.k6 5040bi2 [0,0,0,154083,13653218][0, 0, 0, -154083, 13653218] 135487869158881/51438240000135487869158881/51438240000 153593761628160000153593761628160000 [2,2][2, 2] 4915249152 1.99721.9972  
5040.k5 5040bi3 [0,0,0,1087203,426592798][0, 0, 0, -1087203, -426592798] 47595748626367201/121550625000047595748626367201/1215506250000 36294822144000000003629482214400000000 [2,2][2, 2] 9830498304 2.34372.3437  
5040.k4 5040bi4 [0,0,0,2170083,1230107618][0, 0, 0, -2170083, 1230107618] 378499465220294881/120530818800378499465220294881/120530818800 359903096443699200359903096443699200 [2][2] 9830498304 2.34372.3437  
5040.k2 5040bi5 [0,0,0,17287203,27665272798][0, 0, 0, -17287203, -27665272798] 191342053882402567201/129708022500191342053882402567201/129708022500 387306079856640000387306079856640000 [2,2][2, 2] 196608196608 2.69032.6903  
5040.k8 5040bi6 [0,0,0,182877,1363657822][0, 0, 0, 182877, -1363657822] 226523624554079/269165039062500226523624554079/269165039062500 803722500000000000000-803722500000000000000 [2][2] 196608196608 2.69032.6903  
5040.k1 5040bi7 [0,0,0,276595203,1770578063998][0, 0, 0, -276595203, -1770578063998] 783736670177727068275201/360150783736670177727068275201/360150 10754021376001075402137600 [2][2] 393216393216 3.03693.0369  
5040.k3 5040bi8 [0,0,0,17179203,28028001598][0, 0, 0, -17179203, -28028001598] 187778242790732059201/4984939585440150-187778242790732059201/4984939585440150 14884949843090920857600-14884949843090920857600 [2][2] 393216393216 3.03693.0369  

Rank

sage: E.rank()
 

The elliptic curves in class 5040bi have rank 00.

Complex multiplication

The elliptic curves in class 5040bi do not have complex multiplication.

Modular form 5040.2.a.bi

sage: E.q_eigenform(10)
 
qq5+q74q112q132q174q19+O(q20)q - q^{5} + q^{7} - 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.

(124488161621224488421422444241881616842814228428418816841628141684162841)\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 2 & 2 & 4 & 4 \\ 4 & 2 & 4 & 1 & 8 & 8 & 16 & 16 \\ 8 & 4 & 2 & 8 & 1 & 4 & 2 & 2 \\ 8 & 4 & 2 & 8 & 4 & 1 & 8 & 8 \\ 16 & 8 & 4 & 16 & 2 & 8 & 1 & 4 \\ 16 & 8 & 4 & 16 & 2 & 8 & 4 & 1 \end{array}\right)

Isogeny graph

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

The vertices are labelled with Cremona labels.