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Properties

Label 169338b
Number of curves $2$
Conductor $169338$
CM no
Rank $1$
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Show commands: SageMath
E = EllipticCurve("b1")
 
E.isogeny_class()
 

Elliptic curves in class 169338b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
169338.w2 169338b1 \([1, 0, 0, -764, -33840]\) \(-10218313/96192\) \(-464300411328\) \([2]\) \(338688\) \(0.92105\) \(\Gamma_0(N)\)-optimal
169338.w1 169338b2 \([1, 0, 0, -21044, -1173576]\) \(213525509833/669336\) \(3230757028824\) \([2]\) \(677376\) \(1.2676\)  

Rank

sage: E.rank()
 

The elliptic curves in class 169338b have rank \(1\).

Complex multiplication

The elliptic curves in class 169338b do not have complex multiplication.

Modular form 169338.2.a.b

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{3} + q^{4} - 2 q^{5} + q^{6} + 4 q^{7} + q^{8} + q^{9} - 2 q^{10} + 4 q^{11} + q^{12} + 4 q^{14} - 2 q^{15} + q^{16} - 4 q^{17} + q^{18} + 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.