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Properties

Label 36414cp
Number of curves $2$
Conductor $36414$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("cp1")
 
E.isogeny_class()
 

Elliptic curves in class 36414cp

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
36414.db2 36414cp1 \([1, -1, 1, 81877, 1729019]\) \(3449795831/2071552\) \(-36451625186737152\) \([2]\) \(552960\) \(1.8672\) \(\Gamma_0(N)\)-optimal
36414.db1 36414cp2 \([1, -1, 1, -334283, 14213819]\) \(234770924809/130960928\) \(2304426179774039328\) \([2]\) \(1105920\) \(2.2137\)  

Rank

sage: E.rank()
 

The elliptic curves in class 36414cp have rank \(1\).

Complex multiplication

The elliptic curves in class 36414cp do not have complex multiplication.

Modular form 36414.2.a.cp

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