Properties

Label 32340.z
Number of curves $2$
Conductor $32340$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("z1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 32340.z

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
32340.z1 32340v2 \([0, 1, 0, -108061, -13708465]\) \(227040091070464/4492125\) \(2761111584000\) \([]\) \(139968\) \(1.5086\)  
32340.z2 32340v1 \([0, 1, 0, -2221, 8399]\) \(1972117504/1082565\) \(665405072640\) \([3]\) \(46656\) \(0.95930\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 32340.z have rank \(1\).

Complex multiplication

The elliptic curves in class 32340.z do not have complex multiplication.

Modular form 32340.2.a.z

sage: E.q_eigenform(10)
 
\(q + q^{3} - q^{5} + q^{9} - q^{11} - q^{13} - q^{15} + 2q^{19} + O(q^{20})\)  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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.