Properties

Label 13680.a
Number of curves $4$
Conductor $13680$
CM no
Rank $2$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("13680.a1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 13680.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
13680.a1 13680bh4 [0, 0, 0, -437763, 111482498] [2] 73728  
13680.a2 13680bh3 [0, 0, 0, -31683, 1154882] [2] 73728  
13680.a3 13680bh2 [0, 0, 0, -27363, 1741538] [2, 2] 36864  
13680.a4 13680bh1 [0, 0, 0, -1443, 36002] [2] 18432 \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 13680.a have rank \(2\).

Modular form 13680.2.a.a

sage: E.q_eigenform(10)
 
\( q - q^{5} - 4q^{7} - 4q^{11} - 2q^{13} + 2q^{17} + q^{19} + O(q^{20}) \)

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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.