Properties

Label 39984.e
Number of curves $2$
Conductor $39984$
CM no
Rank $1$
Graph

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

Elliptic curves in class 39984.e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
39984.e1 39984bj2 \([0, -1, 0, -10738072, 13436333296]\) \(5799070911693913/54760833024\) \(1293047009188201365504\) \([]\) \(2177280\) \(2.8717\)  
39984.e2 39984bj1 \([0, -1, 0, -941992, -340873616]\) \(3914907891433/135834624\) \(3207412024401199104\) \([]\) \(725760\) \(2.3224\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 39984.e have rank \(1\).

Complex multiplication

The elliptic curves in class 39984.e do not have complex multiplication.

Modular form 39984.2.a.e

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

\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.