Properties

Label 212160.x
Number of curves $6$
Conductor $212160$
CM no
Rank $2$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("x1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 212160.x

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
212160.x1 212160hl6 \([0, -1, 0, -777921, -261791295]\) \(397210600760070242/3536192675535\) \(463495846367723520\) \([2]\) \(2883584\) \(2.2129\)  
212160.x2 212160hl4 \([0, -1, 0, -84321, 2747745]\) \(1011710313226084/536724738225\) \(35174792444313600\) \([2, 2]\) \(1441792\) \(1.8664\)  
212160.x3 212160hl2 \([0, -1, 0, -66321, 6588945]\) \(1969080716416336/2472575625\) \(40510679040000\) \([2, 2]\) \(720896\) \(1.5198\)  
212160.x4 212160hl1 \([0, -1, 0, -66301, 6593101]\) \(31476797652269056/49725\) \(50918400\) \([2]\) \(360448\) \(1.1732\) \(\Gamma_0(N)\)-optimal
212160.x5 212160hl3 \([0, -1, 0, -48641, 10163841]\) \(-194204905090564/566398828125\) \(-37119513600000000\) \([2]\) \(1441792\) \(1.8664\)  
212160.x6 212160hl5 \([0, -1, 0, 321279, 21161985]\) \(27980756504588158/17683545112935\) \(-2317817625042616320\) \([2]\) \(2883584\) \(2.2129\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 212160.2.a.x

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.