Properties

Label 76608.by
Number of curves $4$
Conductor $76608$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("by1")
 
E.isogeny_class()
 

Elliptic curves in class 76608.by

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
76608.by1 76608dh3 \([0, 0, 0, -9739980, -8339164272]\) \(19804628171203875/5638671302656\) \(29094305398830674214912\) \([2]\) \(5308416\) \(3.0171\)  
76608.by2 76608dh1 \([0, 0, 0, -8940300, -10289072624]\) \(11165451838341046875/572244736\) \(4050284149997568\) \([2]\) \(1769472\) \(2.4678\) \(\Gamma_0(N)\)-optimal
76608.by3 76608dh2 \([0, 0, 0, -8924940, -10326188528]\) \(-11108001800138902875/79947274872976\) \(-565857857456138354688\) \([2]\) \(3538944\) \(2.8143\)  
76608.by4 76608dh4 \([0, 0, 0, 25649460, -55010757744]\) \(361682234074684125/462672528510976\) \(-2387288622021093781143552\) \([2]\) \(10616832\) \(3.3636\)  

Rank

sage: E.rank()
 

The elliptic curves in class 76608.by have rank \(1\).

Complex multiplication

The elliptic curves in class 76608.by do not have complex multiplication.

Modular form 76608.2.a.by

sage: E.q_eigenform(10)
 
\(q - q^{7} - 6 q^{11} - 2 q^{13} + q^{19} + O(q^{20})\) 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 LMFDB numbering.

\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.