Properties

Label 182070.bi
Number of curves $8$
Conductor $182070$
CM no
Rank $2$
Graph

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

Elliptic curves in class 182070.bi

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
182070.bi1 182070cj7 \([1, -1, 0, -4996000854, 135920780694310]\) \(783736670177727068275201/360150\) \(6337303051530150\) \([2]\) \(83886080\) \(3.7603\)  
182070.bi2 182070cj5 \([1, -1, 0, -312250104, 2123820019660]\) \(191342053882402567201/129708022500\) \(2282379694008583522500\) \([2, 2]\) \(41943040\) \(3.4138\)  
182070.bi3 182070cj8 \([1, -1, 0, -310299354, 2151664635010]\) \(-187778242790732059201/4984939585440150\) \(-87716431616002508660610150\) \([2]\) \(83886080\) \(3.7603\)  
182070.bi4 182070cj4 \([1, -1, 0, -39197124, -94420180832]\) \(378499465220294881/120530818800\) \(2120894976494981458800\) \([2]\) \(20971520\) \(3.0672\)  
182070.bi5 182070cj3 \([1, -1, 0, -19637604, 32752572160]\) \(47595748626367201/1215506250000\) \(21388397798914256250000\) \([2, 2]\) \(20971520\) \(3.0672\)  
182070.bi6 182070cj2 \([1, -1, 0, -2783124, -1047402032]\) \(135487869158881/51438240000\) \(905122075016910240000\) \([2, 2]\) \(10485760\) \(2.7206\)  
182070.bi7 182070cj1 \([1, -1, 0, 546156, -117201200]\) \(1023887723039/928972800\) \(-16346472748100812800\) \([2]\) \(5242880\) \(2.3740\) \(\Gamma_0(N)\)-optimal
182070.bi8 182070cj6 \([1, -1, 0, 3303216, 104681219188]\) \(226523624554079/269165039062500\) \(-4736305493311157226562500\) \([2]\) \(41943040\) \(3.4138\)  

Rank

sage: E.rank()
 

The elliptic curves in class 182070.bi have rank \(2\).

Complex multiplication

The elliptic curves in class 182070.bi do not have complex multiplication.

Modular form 182070.2.a.bi

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{4} + q^{5} + q^{7} - q^{8} - q^{10} - 4 q^{11} - 2 q^{13} - q^{14} + q^{16} + 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

sage: E.isogeny_class().matrix()
 

The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.