Properties

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

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Show commands: SageMath
E = EllipticCurve("j1")
 
E.isogeny_class()
 

Elliptic curves in class 13680.j

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
13680.j1 13680bc4 \([0, 0, 0, -1076763, -415287862]\) \(46237740924063961/1806561830400\) \(5394364720585113600\) \([2]\) \(165888\) \(2.3616\)  
13680.j2 13680bc2 \([0, 0, 0, -158763, 24174938]\) \(148212258825961/1218375000\) \(3638048256000000\) \([2]\) \(55296\) \(1.8122\)  
13680.j3 13680bc1 \([0, 0, 0, -3243, 878042]\) \(-1263214441/110808000\) \(-330870915072000\) \([2]\) \(27648\) \(1.4657\) \(\Gamma_0(N)\)-optimal
13680.j4 13680bc3 \([0, 0, 0, 29157, -23570998]\) \(918046641959/80912056320\) \(-241602105578618880\) \([2]\) \(82944\) \(2.0150\)  

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 13680.2.a.j

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

\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.