Properties

Label 13860.f
Number of curves $4$
Conductor $13860$
CM no
Rank $0$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("f1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 13860.f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
13860.f1 13860o4 \([0, 0, 0, -1765263, 901827862]\) \(3259751350395879376/3806353980275\) \(710357005214841600\) \([6]\) \(248832\) \(2.3375\)  
13860.f2 13860o3 \([0, 0, 0, -1764768, 902359393]\) \(52112158467655991296/71177645\) \(830216051280\) \([6]\) \(124416\) \(1.9909\)  
13860.f3 13860o2 \([0, 0, 0, -82263, -7803938]\) \(329890530231376/49933296875\) \(9318751596000000\) \([2]\) \(82944\) \(1.7882\)  
13860.f4 13860o1 \([0, 0, 0, -22368, 1168333]\) \(106110329552896/10850811125\) \(126563860962000\) \([2]\) \(41472\) \(1.4416\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 13860.f have rank \(0\).

Complex multiplication

The elliptic curves in class 13860.f do not have complex multiplication.

Modular form 13860.2.a.f

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.