Properties

Label 169065x
Number of curves $8$
Conductor $169065$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("x1")
 
E.isogeny_class()
 

Elliptic curves in class 169065x

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
169065.bi6 169065x1 \([1, -1, 0, -286164, -58847765]\) \(147281603041/5265\) \(92644455272265\) \([2]\) \(983040\) \(1.7690\) \(\Gamma_0(N)\)-optimal
169065.bi5 169065x2 \([1, -1, 0, -299169, -53195792]\) \(168288035761/27720225\) \(487773057008475225\) \([2, 2]\) \(1966080\) \(2.1156\)  
169065.bi4 169065x3 \([1, -1, 0, -1352574, 555040255]\) \(15551989015681/1445900625\) \(25442483529145775625\) \([2, 2]\) \(3932160\) \(2.4622\)  
169065.bi7 169065x4 \([1, -1, 0, 546156, -299861627]\) \(1023887723039/2798036865\) \(-49235061954347783865\) \([2]\) \(3932160\) \(2.4622\)  
169065.bi2 169065x5 \([1, -1, 0, -21133179, 37398395128]\) \(59319456301170001/594140625\) \(10454669431766015625\) \([2, 2]\) \(7864320\) \(2.8088\)  
169065.bi8 169065x6 \([1, -1, 0, 1573551, 2626151530]\) \(24487529386319/183539412225\) \(-3229612320337477767225\) \([2]\) \(7864320\) \(2.8088\)  
169065.bi1 169065x7 \([1, -1, 0, -338130054, 2393255770753]\) \(242970740812818720001/24375\) \(428909515149375\) \([2]\) \(15728640\) \(3.1553\)  
169065.bi3 169065x8 \([1, -1, 0, -20625984, 39278364115]\) \(-55150149867714721/5950927734375\) \(-104714237097015380859375\) \([2]\) \(15728640\) \(3.1553\)  

Rank

sage: E.rank()
 

The elliptic curves in class 169065x have rank \(0\).

Complex multiplication

The elliptic curves in class 169065x do not have complex multiplication.

Modular form 169065.2.a.x

sage: E.q_eigenform(10)
 
\(q + q^{2} - q^{4} + q^{5} - 3 q^{8} + q^{10} + 4 q^{11} + q^{13} - q^{16} - 4 q^{19} + O(q^{20})\) 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 Cremona numbering.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.