Properties

Label 55440er
Number of curves $4$
Conductor $55440$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("er1")
 
E.isogeny_class()
 

Elliptic curves in class 55440er

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
55440.dw3 55440er1 \([0, 0, 0, -142707, -14301646]\) \(107639597521009/32699842560\) \(97641206686679040\) \([2]\) \(491520\) \(1.9650\) \(\Gamma_0(N)\)-optimal
55440.dw2 55440er2 \([0, 0, 0, -879987, 306710066]\) \(25238585142450289/995844326400\) \(2973575225121177600\) \([2, 2]\) \(983040\) \(2.3116\)  
55440.dw4 55440er3 \([0, 0, 0, 387213, 1117464626]\) \(2150235484224911/181905111732960\) \(-543165753152830832640\) \([2]\) \(1966080\) \(2.6581\)  
55440.dw1 55440er4 \([0, 0, 0, -13943667, 20040705074]\) \(100407751863770656369/166028940000\) \(495759758376960000\) \([2]\) \(1966080\) \(2.6581\)  

Rank

sage: E.rank()
 

The elliptic curves in class 55440er have rank \(1\).

Complex multiplication

The elliptic curves in class 55440er do not have complex multiplication.

Modular form 55440.2.a.er

sage: E.q_eigenform(10)
 
\(q + q^{5} + q^{7} - q^{11} - 6 q^{13} + 6 q^{17} + 4 q^{19} + O(q^{20})\) 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 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.