Properties

Label 171.b
Number of curves $3$
Conductor $171$
CM no
Rank $1$
Graph

Related objects

Downloads

Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("b1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 171.b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
171.b1 171b3 \([0, 0, 1, -6924, 221760]\) \(-50357871050752/19\) \(-13851\) \([3]\) \(72\) \(0.58275\)  
171.b2 171b2 \([0, 0, 1, -84, 315]\) \(-89915392/6859\) \(-5000211\) \([3]\) \(24\) \(0.033439\)  
171.b3 171b1 \([0, 0, 1, 6, 0]\) \(32768/19\) \(-13851\) \([]\) \(8\) \(-0.51587\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 171.b have rank \(1\).

Complex multiplication

The elliptic curves in class 171.b do not have complex multiplication.

Modular form 171.2.a.b

sage: E.q_eigenform(10)
 
\(q - 2q^{4} - 3q^{5} - q^{7} - 3q^{11} - 4q^{13} + 4q^{16} + 3q^{17} + q^{19} + O(q^{20})\)  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.

\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.