Properties

Label 82368.co
Number of curves $2$
Conductor $82368$
CM no
Rank $1$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
E = EllipticCurve("co1")
 
E.isogeny_class()
 

Elliptic curves in class 82368.co

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
82368.co1 82368dj2 \([0, 0, 0, -7500, 188048]\) \(244140625/61347\) \(11723594268672\) \([2]\) \(131072\) \(1.2177\)  
82368.co2 82368dj1 \([0, 0, 0, 1140, 18704]\) \(857375/1287\) \(-245949530112\) \([2]\) \(65536\) \(0.87109\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 82368.co have rank \(1\).

Complex multiplication

The elliptic curves in class 82368.co do not have complex multiplication.

Modular form 82368.2.a.co

sage: E.q_eigenform(10)
 
\(q - q^{11} - q^{13} + 4 q^{17} - 8 q^{19} + O(q^{20})\) 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.