Properties

Label 277200.gy
Number of curves $6$
Conductor $277200$
CM no
Rank $2$
Graph

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Show commands: SageMath
E = EllipticCurve("gy1")
 
E.isogeny_class()
 

Elliptic curves in class 277200.gy

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
277200.gy1 277200gy6 \([0, 0, 0, -16268475, -25256209750]\) \(10206027697760497/5557167\) \(259275183552000000\) \([2]\) \(10485760\) \(2.6700\)  
277200.gy2 277200gy4 \([0, 0, 0, -1022475, -389983750]\) \(2533811507137/58110129\) \(2711186178624000000\) \([2, 2]\) \(5242880\) \(2.3234\)  
277200.gy3 277200gy2 \([0, 0, 0, -140475, 11326250]\) \(6570725617/2614689\) \(121990929984000000\) \([2, 2]\) \(2621440\) \(1.9768\)  
277200.gy4 277200gy1 \([0, 0, 0, -122475, 16492250]\) \(4354703137/1617\) \(75442752000000\) \([2]\) \(1310720\) \(1.6302\) \(\Gamma_0(N)\)-optimal
277200.gy5 277200gy5 \([0, 0, 0, 111525, -1207597750]\) \(3288008303/13504609503\) \(-630071060971968000000\) \([2]\) \(10485760\) \(2.6700\)  
277200.gy6 277200gy3 \([0, 0, 0, 453525, 82012250]\) \(221115865823/190238433\) \(-8875764330048000000\) \([2]\) \(5242880\) \(2.3234\)  

Rank

sage: E.rank()
 

The elliptic curves in class 277200.gy have rank \(2\).

Complex multiplication

The elliptic curves in class 277200.gy do not have complex multiplication.

Modular form 277200.2.a.gy

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.