Properties

Label 13680.bk
Number of curves $4$
Conductor $13680$
CM no
Rank $1$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("bk1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 13680.bk

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
13680.bk1 13680br3 \([0, 0, 0, -89427, -10278254]\) \(26487576322129/44531250\) \(132969600000000\) \([2]\) \(49152\) \(1.6054\)  
13680.bk2 13680br2 \([0, 0, 0, -7347, -51086]\) \(14688124849/8122500\) \(24253655040000\) \([2, 2]\) \(24576\) \(1.2588\)  
13680.bk3 13680br1 \([0, 0, 0, -4467, 114226]\) \(3301293169/22800\) \(68080435200\) \([2]\) \(12288\) \(0.91222\) \(\Gamma_0(N)\)-optimal
13680.bk4 13680br4 \([0, 0, 0, 28653, -403886]\) \(871257511151/527800050\) \(-1576002504499200\) \([2]\) \(49152\) \(1.6054\)  

Rank

sage: E.rank()
 

The elliptic curves in class 13680.bk have rank \(1\).

Complex multiplication

The elliptic curves in class 13680.bk do not have complex multiplication.

Modular form 13680.2.a.bk

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