Properties

Label 123786bn
Number of curves $4$
Conductor $123786$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("bn1")
 
E.isogeny_class()
 

Elliptic curves in class 123786bn

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
123786.bm4 123786bn1 \([1, -1, 1, -92939, 227700123]\) \(-822656953/207028224\) \(-22342105640001798144\) \([2]\) \(4055040\) \(2.3920\) \(\Gamma_0(N)\)-optimal
123786.bm3 123786bn2 \([1, -1, 1, -6187019, 5870818203]\) \(242702053576633/2554695936\) \(275698092643615938816\) \([2, 2]\) \(8110080\) \(2.7386\)  
123786.bm2 123786bn3 \([1, -1, 1, -11138459, -4850039685]\) \(1416134368422073/725251155408\) \(78267772464004219092048\) \([2]\) \(16220160\) \(3.0851\)  
123786.bm1 123786bn4 \([1, -1, 1, -98740859, 377678104251]\) \(986551739719628473/111045168\) \(11983790549581042608\) \([2]\) \(16220160\) \(3.0851\)  

Rank

sage: E.rank()
 

The elliptic curves in class 123786bn have rank \(1\).

Complex multiplication

The elliptic curves in class 123786bn do not have complex multiplication.

Modular form 123786.2.a.bn

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

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.