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Properties

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

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Show commands: SageMath
E = EllipticCurve("u1")
 
E.isogeny_class()
 

Elliptic curves in class 82110.u

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
82110.u1 82110s2 \([1, 0, 1, -142605354, -648136131428]\) \(320723808872496523935142301209/4135958612138279753487360\) \(4135958612138279753487360\) \([2]\) \(23961600\) \(3.5318\)  
82110.u2 82110s1 \([1, 0, 1, -16832554, 10611485852]\) \(527440803339012896847466009/256574315980928345702400\) \(256574315980928345702400\) \([2]\) \(11980800\) \(3.1853\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 82110.u have rank \(2\).

Complex multiplication

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

Modular form 82110.2.a.u

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.