Properties

Label 55440.e
Number of curves $6$
Conductor $55440$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("e1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 55440.e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
55440.e1 55440ct6 [0, 0, 0, -439084803, -3541365261502] [2] 3932160  
55440.e2 55440ct4 [0, 0, 0, -27442803, -55333820302] [2, 2] 1966080  
55440.e3 55440ct5 [0, 0, 0, -27306723, -55909738078] [2] 3932160  
55440.e4 55440ct3 [0, 0, 0, -3664083, 1423613522] [2] 1966080  
55440.e5 55440ct2 [0, 0, 0, -1723683, -855580318] [2, 2] 983040  
55440.e6 55440ct1 [0, 0, 0, 5037, -39970222] [2] 491520 \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 55440.2.a.e

sage: E.q_eigenform(10)
 
\( q - q^{5} - q^{7} - q^{11} - 2q^{13} - 2q^{17} - 4q^{19} + O(q^{20}) \)

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.