Properties

Label 16170cf
Number of curves 4
Conductor 16170
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("16170.cd1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 16170cf

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
16170.cd3 16170cf1 [1, 0, 0, -24550, 1283540] [4] 73728 \(\Gamma_0(N)\)-optimal
16170.cd2 16170cf2 [1, 0, 0, -103930, -11623648] [2, 2] 147456  
16170.cd1 16170cf3 [1, 0, 0, -1616560, -791233150] [2] 294912  
16170.cd4 16170cf4 [1, 0, 0, 138620, -57756658] [2] 294912  

Rank

sage: E.rank()
 

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

Modular form 16170.2.a.cd

sage: E.q_eigenform(10)
 
\( q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} + q^{8} + q^{9} + q^{10} - q^{11} + q^{12} - 2q^{13} + q^{15} + q^{16} - 2q^{17} + q^{18} - 4q^{19} + O(q^{20}) \)

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.