Properties

Label 212160.cx
Number of curves $4$
Conductor $212160$
CM no
Rank $2$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("cx1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 212160.cx

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
212160.cx1 212160gl4 \([0, -1, 0, -130785, -18161055]\) \(1887517194957938/21849165\) \(2863813754880\) \([2]\) \(786432\) \(1.5414\)  
212160.cx2 212160gl2 \([0, -1, 0, -8385, -266175]\) \(994958062276/98903025\) \(6481708646400\) \([2, 2]\) \(393216\) \(1.1949\)  
212160.cx3 212160gl1 \([0, -1, 0, -1905, 28017]\) \(46689225424/7249905\) \(118782443520\) \([2]\) \(196608\) \(0.84829\) \(\Gamma_0(N)\)-optimal
212160.cx4 212160gl3 \([0, -1, 0, 10335, -1303263]\) \(931329171502/6107473125\) \(-800518717440000\) \([4]\) \(786432\) \(1.5414\)  

Rank

sage: E.rank()
 

The elliptic curves in class 212160.cx have rank \(2\).

Complex multiplication

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

Modular form 212160.2.a.cx

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