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Properties

Label 16170.bc
Number of curves $4$
Conductor $16170$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("bc1")
 
E.isogeny_class()
 

Elliptic curves in class 16170.bc

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
16170.bc1 16170bh3 \([1, 0, 1, -146546188, -680360489014]\) \(2958414657792917260183849/12401051653985258880\) \(1458971326039711721973120\) \([2]\) \(4816896\) \(3.4913\)  
16170.bc2 16170bh2 \([1, 0, 1, -13736588, 1112130506]\) \(2436531580079063806249/1405478914998681600\) \(165353188870679891558400\) \([2, 2]\) \(2408448\) \(3.1447\)  
16170.bc3 16170bh1 \([1, 0, 1, -9722508, 11638653898]\) \(863913648706111516969/2486234429521920\) \(292502994398824366080\) \([2]\) \(1204224\) \(2.7982\) \(\Gamma_0(N)\)-optimal
16170.bc4 16170bh4 \([1, 0, 1, 54847732, 8903309258]\) \(155099895405729262880471/90047655797243760000\) \(-10594016656889931120240000\) \([2]\) \(4816896\) \(3.4913\)  

Rank

sage: E.rank()
 

The elliptic curves in class 16170.bc have rank \(1\).

Complex multiplication

The elliptic curves in class 16170.bc do not have complex multiplication.

Modular form 16170.2.a.bc

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} - q^{8} + q^{9} - q^{10} + q^{11} + q^{12} - 6 q^{13} + q^{15} + q^{16} - 2 q^{17} - q^{18} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.