Properties

Label 122694be
Number of curves $4$
Conductor $122694$
CM no
Rank $1$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
E = EllipticCurve("be1")
 
E.isogeny_class()
 

Elliptic curves in class 122694be

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
122694.bh4 122694be1 \([1, 0, 1, 7350989, -5515298098]\) \(5137417856375/4510142208\) \(-38566165489308394958592\) \([2]\) \(11612160\) \(3.0214\) \(\Gamma_0(N)\)-optimal
122694.bh3 122694be2 \([1, 0, 1, -36818851, -48996088594]\) \(645532578015625/252306960048\) \(2157473429120638844824752\) \([2]\) \(23224320\) \(3.3679\)  
122694.bh2 122694be3 \([1, 0, 1, -76387666, 343138965860]\) \(-5764706497797625/2612665516032\) \(-22340867762611228841607168\) \([2]\) \(34836480\) \(3.5707\)  
122694.bh1 122694be4 \([1, 0, 1, -1332774226, 18725582002532]\) \(30618029936661765625/3678951124992\) \(31458661694048021862494208\) \([2]\) \(69672960\) \(3.9172\)  

Rank

sage: E.rank()
 

The elliptic curves in class 122694be have rank \(1\).

Complex multiplication

The elliptic curves in class 122694be do not have complex multiplication.

Modular form 122694.2.a.be

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{3} + q^{4} - q^{6} - 4 q^{7} - q^{8} + q^{9} + q^{12} + 4 q^{14} + q^{16} - q^{18} - 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.