Properties

Label 277440.ge
Number of curves $4$
Conductor $277440$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("ge1")
 
E.isogeny_class()
 

Elliptic curves in class 277440.ge

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
277440.ge1 277440ge4 \([0, 1, 0, -65121, -6359841]\) \(38614472/405\) \(320330643701760\) \([2]\) \(1310720\) \(1.5999\)  
277440.ge2 277440ge2 \([0, 1, 0, -7321, 79079]\) \(438976/225\) \(22245183590400\) \([2, 2]\) \(655360\) \(1.2533\)  
277440.ge3 277440ge1 \([0, 1, 0, -5876, 171270]\) \(14526784/15\) \(23172066240\) \([2]\) \(327680\) \(0.90676\) \(\Gamma_0(N)\)-optimal
277440.ge4 277440ge3 \([0, 1, 0, 27359, 640895]\) \(2863288/1875\) \(-1483012239360000\) \([2]\) \(1310720\) \(1.5999\)  

Rank

sage: E.rank()
 

The elliptic curves in class 277440.ge have rank \(1\).

Complex multiplication

The elliptic curves in class 277440.ge do not have complex multiplication.

Modular form 277440.2.a.ge

sage: E.q_eigenform(10)
 
\(q + q^{3} - q^{5} + q^{9} + 4 q^{11} - 2 q^{13} - q^{15} - 8 q^{19} + O(q^{20})\) 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 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.