Properties

Label 15680.bw
Number of curves $4$
Conductor $15680$
CM no
Rank $2$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("bw1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 15680.bw

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
15680.bw1 15680e3 \([0, 0, 0, -16268, 773808]\) \(123505992/4375\) \(16866160640000\) \([2]\) \(36864\) \(1.3087\)  
15680.bw2 15680e2 \([0, 0, 0, -2548, -32928]\) \(3796416/1225\) \(590315622400\) \([2, 2]\) \(18432\) \(0.96212\)  
15680.bw3 15680e1 \([0, 0, 0, -2303, -42532]\) \(179406144/35\) \(263533760\) \([2]\) \(9216\) \(0.61555\) \(\Gamma_0(N)\)-optimal
15680.bw4 15680e4 \([0, 0, 0, 7252, -225008]\) \(10941048/12005\) \(-46280744796160\) \([2]\) \(36864\) \(1.3087\)  

Rank

sage: E.rank()
 

The elliptic curves in class 15680.bw have rank \(2\).

Complex multiplication

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

Modular form 15680.2.a.bw

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.