Properties

Label 212160.gy
Number of curves $4$
Conductor $212160$
CM no
Rank $0$
Graph

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

Elliptic curves in class 212160.gy

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
212160.gy1 212160q3 \([0, 1, 0, -130785, 18161055]\) \(1887517194957938/21849165\) \(2863813754880\) \([4]\) \(786432\) \(1.5414\)  
212160.gy2 212160q2 \([0, 1, 0, -8385, 266175]\) \(994958062276/98903025\) \(6481708646400\) \([2, 2]\) \(393216\) \(1.1949\)  
212160.gy3 212160q1 \([0, 1, 0, -1905, -28017]\) \(46689225424/7249905\) \(118782443520\) \([2]\) \(196608\) \(0.84829\) \(\Gamma_0(N)\)-optimal
212160.gy4 212160q4 \([0, 1, 0, 10335, 1303263]\) \(931329171502/6107473125\) \(-800518717440000\) \([2]\) \(786432\) \(1.5414\)  

Rank

sage: E.rank()
 

The elliptic curves in class 212160.gy have rank \(0\).

Complex multiplication

The elliptic curves in class 212160.gy do not have complex multiplication.

Modular form 212160.2.a.gy

sage: E.q_eigenform(10)
 
\(q + q^{3} + q^{5} + q^{9} + 4q^{11} + q^{13} + q^{15} + q^{17} + 8q^{19} + O(q^{20})\)  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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.