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Properties

Label 82038.e
Number of curves $2$
Conductor $82038$
CM no
Rank $2$
Graph

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

Elliptic curves in class 82038.e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
82038.e1 82038e2 \([1, 0, 1, -861644, 3416351750]\) \(-39934705050538129/2823126576537804\) \(-5001340941057888592044\) \([]\) \(4939200\) \(2.8434\)  
82038.e2 82038e1 \([1, 0, 1, -200984, -34936090]\) \(-506814405937489/4048994304\) \(-7173040398188544\) \([]\) \(705600\) \(1.8704\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 82038.e have rank \(2\).

Complex multiplication

The elliptic curves in class 82038.e do not have complex multiplication.

Modular form 82038.2.a.e

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} - q^{7} - q^{8} + q^{9} + q^{10} + q^{12} - 7 q^{13} + q^{14} - q^{15} + q^{16} + 3 q^{17} - q^{18} - 6 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.