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Properties

Label 86190d
Number of curves $8$
Conductor $86190$
CM no
Rank $2$
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Show commands: SageMath
E = EllipticCurve("d1")
 
E.isogeny_class()
 

Elliptic curves in class 86190d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
86190.d6 86190d1 \([1, 1, 0, -13523, 738093]\) \(-56667352321/16711680\) \(-80664087429120\) \([2]\) \(294912\) \(1.3829\) \(\Gamma_0(N)\)-optimal
86190.d5 86190d2 \([1, 1, 0, -229843, 42314797]\) \(278202094583041/16646400\) \(80348993337600\) \([2, 2]\) \(589824\) \(1.7295\)  
86190.d4 86190d3 \([1, 1, 0, -243363, 37039293]\) \(330240275458561/67652010000\) \(326543330736090000\) \([2, 2]\) \(1179648\) \(2.0761\)  
86190.d2 86190d4 \([1, 1, 0, -3677443, 2712825757]\) \(1139466686381936641/4080\) \(19693380720\) \([2]\) \(1179648\) \(2.0761\)  
86190.d7 86190d5 \([1, 1, 0, 517137, 223057593]\) \(3168685387909439/6278181696900\) \(-30303583918232192100\) \([2]\) \(2359296\) \(2.4226\)  
86190.d3 86190d6 \([1, 1, 0, -1220183, -486340863]\) \(41623544884956481/2962701562500\) \(14300394566189062500\) \([2, 2]\) \(2359296\) \(2.4226\)  
86190.d8 86190d7 \([1, 1, 0, 1106947, -2119520697]\) \(31077313442863199/420227050781250\) \(-2028355710754394531250\) \([2]\) \(4718592\) \(2.7692\)  
86190.d1 86190d8 \([1, 1, 0, -19176433, -32329954613]\) \(161572377633716256481/914742821250\) \(4415288882294891250\) \([2]\) \(4718592\) \(2.7692\)  

Rank

sage: E.rank()
 

The elliptic curves in class 86190d have rank \(2\).

Complex multiplication

The elliptic curves in class 86190d do not have complex multiplication.

Modular form 86190.2.a.d

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{3} + q^{4} - q^{5} + q^{6} - q^{8} + q^{9} + q^{10} - 4 q^{11} - q^{12} + q^{15} + q^{16} + q^{17} - q^{18} - 4 q^{19} + O(q^{20})\) 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 Cremona numbering.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.