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Properties

Label 82110.x
Number of curves $2$
Conductor $82110$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("x1")
 
E.isogeny_class()
 

Elliptic curves in class 82110.x

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
82110.x1 82110z2 \([1, 0, 1, -20289, 1093786]\) \(923580740393079049/16049455024050\) \(16049455024050\) \([2]\) \(270336\) \(1.3307\)  
82110.x2 82110z1 \([1, 0, 1, -39, 48886]\) \(-6321363049/1032553777500\) \(-1032553777500\) \([2]\) \(135168\) \(0.98414\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 82110.x have rank \(1\).

Complex multiplication

The elliptic curves in class 82110.x do not have complex multiplication.

Modular form 82110.2.a.x

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} + q^{7} - q^{8} + q^{9} + q^{10} - 2 q^{11} + q^{12} + 4 q^{13} - q^{14} - q^{15} + q^{16} + q^{17} - q^{18} + 2 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.