Properties

Label 22386ba
Number of curves $4$
Conductor $22386$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("ba1")
 
E.isogeny_class()
 

Elliptic curves in class 22386ba

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
22386.ba3 22386ba1 \([1, 0, 0, -643, 17153]\) \(-29403487464625/110884842432\) \(-110884842432\) \([6]\) \(20736\) \(0.80494\) \(\Gamma_0(N)\)-optimal
22386.ba2 22386ba2 \([1, 0, 0, -14683, 682649]\) \(350082141630936625/555332456952\) \(555332456952\) \([6]\) \(41472\) \(1.1515\)  
22386.ba4 22386ba3 \([1, 0, 0, 5657, -408475]\) \(20020616659055375/83832462778428\) \(-83832462778428\) \([2]\) \(62208\) \(1.3542\)  
22386.ba1 22386ba4 \([1, 0, 0, -60253, -5035357]\) \(24191354664255948625/3068177831138238\) \(3068177831138238\) \([2]\) \(124416\) \(1.7008\)  

Rank

sage: E.rank()
 

The elliptic curves in class 22386ba have rank \(1\).

Complex multiplication

The elliptic curves in class 22386ba do not have complex multiplication.

Modular form 22386.2.a.ba

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

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.