Properties

Label 104742.p
Number of curves $4$
Conductor $104742$
CM no
Rank $2$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
E = EllipticCurve("p1")
 
E.isogeny_class()
 

Elliptic curves in class 104742.p

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
104742.p1 104742a4 \([1, -1, 0, -4878537, 4148676557]\) \(4406910829875/7744\) \(22564392882280128\) \([2]\) \(2433024\) \(2.3971\)  
104742.p2 104742a3 \([1, -1, 0, -307977, 63510029]\) \(1108717875/45056\) \(131283740405993472\) \([2]\) \(1216512\) \(2.0505\)  
104742.p3 104742a2 \([1, -1, 0, -77862, 2109528]\) \(13060888875/7086244\) \(28323497615694732\) \([2]\) \(811008\) \(1.8478\)  
104742.p4 104742a1 \([1, -1, 0, -46122, -3775068]\) \(2714704875/21296\) \(85119451887888\) \([2]\) \(405504\) \(1.5012\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 104742.p have rank \(2\).

Complex multiplication

The elliptic curves in class 104742.p do not have complex multiplication.

Modular form 104742.2.a.p

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{4} - 2 q^{7} - q^{8} - q^{11} + 2 q^{13} + 2 q^{14} + q^{16} - 6 q^{17} - 2 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.