Properties

Label 142296cy
Number of curves $4$
Conductor $142296$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("cy1")
 
E.isogeny_class()
 

Elliptic curves in class 142296cy

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
142296.m4 142296cy1 \([0, -1, 0, 164036, -1012306700]\) \(9148592/8301447\) \(-442933079532208328448\) \([2]\) \(5898240\) \(2.6407\) \(\Gamma_0(N)\)-optimal
142296.m3 142296cy2 \([0, -1, 0, -14184144, -20083907556]\) \(1478729816932/38900169\) \(8302249788256599081984\) \([2, 2]\) \(11796480\) \(2.9873\)  
142296.m2 142296cy3 \([0, -1, 0, -32445464, 42260238924]\) \(8849350367426/3314597517\) \(1414832749629910300698624\) \([2]\) \(23592960\) \(3.3338\)  
142296.m1 142296cy4 \([0, -1, 0, -225493704, -1303240079700]\) \(2970658109581346/2139291\) \(913154297698256689152\) \([2]\) \(23592960\) \(3.3338\)  

Rank

sage: E.rank()
 

The elliptic curves in class 142296cy have rank \(1\).

Complex multiplication

The elliptic curves in class 142296cy do not have complex multiplication.

Modular form 142296.2.a.cy

sage: E.q_eigenform(10)
 
\(q - q^{3} - 2 q^{5} + q^{9} + 6 q^{13} + 2 q^{15} + 6 q^{17} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.