Properties

Label 13680bf
Number of curves $4$
Conductor $13680$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("bf1")
 
E.isogeny_class()
 

Elliptic curves in class 13680bf

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
13680.ba4 13680bf1 \([0, 0, 0, 1797, -25558]\) \(214921799/218880\) \(-653572177920\) \([2]\) \(24576\) \(0.95399\) \(\Gamma_0(N)\)-optimal
13680.ba3 13680bf2 \([0, 0, 0, -9723, -235222]\) \(34043726521/11696400\) \(34925263257600\) \([2, 2]\) \(49152\) \(1.3006\)  
13680.ba1 13680bf3 \([0, 0, 0, -139323, -20012182]\) \(100162392144121/23457780\) \(70044555755520\) \([2]\) \(98304\) \(1.6471\)  
13680.ba2 13680bf4 \([0, 0, 0, -64443, 6123242]\) \(9912050027641/311647500\) \(930574448640000\) \([2]\) \(98304\) \(1.6471\)  

Rank

sage: E.rank()
 

The elliptic curves in class 13680bf have rank \(0\).

Complex multiplication

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

Modular form 13680.2.a.bf

sage: E.q_eigenform(10)
 
\(q - q^{5} + 4 q^{7} - 4 q^{11} - 6 q^{13} + 6 q^{17} + q^{19} + O(q^{20})\) 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 Cremona 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 Cremona labels.