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Properties

Label 52022.l
Number of curves $3$
Conductor $52022$
CM no
Rank $2$
Graph

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Show commands: SageMath
E = EllipticCurve("l1")
 
E.isogeny_class()
 

Elliptic curves in class 52022.l

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
52022.l1 52022i3 \([1, 0, 0, -117078, -124039900]\) \(-69173457625/2550136832\) \(-6542953416425996288\) \([]\) \(933120\) \(2.2906\)  
52022.l2 52022i1 \([1, 0, 0, -21248, 1190744]\) \(-413493625/152\) \(-389990414168\) \([]\) \(103680\) \(1.1920\) \(\Gamma_0(N)\)-optimal
52022.l3 52022i2 \([1, 0, 0, 12977, 4532473]\) \(94196375/3511808\) \(-9010338528937472\) \([]\) \(311040\) \(1.7413\)  

Rank

sage: E.rank()
 

The elliptic curves in class 52022.l have rank \(2\).

Complex multiplication

The elliptic curves in class 52022.l do not have complex multiplication.

Modular form 52022.2.a.l

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

\(\left(\begin{array}{rrr} 1 & 9 & 3 \\ 9 & 1 & 3 \\ 3 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.