Properties

Label 15680.by
Number of curves $4$
Conductor $15680$
CM no
Rank $0$
Graph

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

Elliptic curves in class 15680.by

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
15680.by1 15680d3 \([0, 0, 0, -16268, -773808]\) \(123505992/4375\) \(16866160640000\) \([2]\) \(36864\) \(1.3087\)  
15680.by2 15680d2 \([0, 0, 0, -2548, 32928]\) \(3796416/1225\) \(590315622400\) \([2, 2]\) \(18432\) \(0.96212\)  
15680.by3 15680d1 \([0, 0, 0, -2303, 42532]\) \(179406144/35\) \(263533760\) \([2]\) \(9216\) \(0.61555\) \(\Gamma_0(N)\)-optimal
15680.by4 15680d4 \([0, 0, 0, 7252, 225008]\) \(10941048/12005\) \(-46280744796160\) \([2]\) \(36864\) \(1.3087\)  

Rank

sage: E.rank()
 

The elliptic curves in class 15680.by have rank \(0\).

Complex multiplication

The elliptic curves in class 15680.by do not have complex multiplication.

Modular form 15680.2.a.by

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

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.