Properties

Label 13872.f
Number of curves $4$
Conductor $13872$
CM no
Rank $0$
Graph

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

Elliptic curves in class 13872.f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
13872.f1 13872w4 \([0, -1, 0, -97541064, 360095102064]\) \(211293405175481/6973568802\) \(3387313012059784178049024\) \([2]\) \(2611200\) \(3.4800\)  
13872.f2 13872w3 \([0, -1, 0, -96754984, 366349783408]\) \(206226044828441/236196\) \(114728886587741847552\) \([2]\) \(1305600\) \(3.1334\)  
13872.f3 13872w2 \([0, -1, 0, -13430504, -18940096656]\) \(551569744601/2592\) \(1259027562005397504\) \([2]\) \(522240\) \(2.6753\)  
13872.f4 13872w1 \([0, -1, 0, -853224, -285474960]\) \(141420761/9216\) \(4476542442685857792\) \([2]\) \(261120\) \(2.3287\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 13872.f have rank \(0\).

Complex multiplication

The elliptic curves in class 13872.f do not have complex multiplication.

Modular form 13872.2.a.f

sage: E.q_eigenform(10)
 
\(q - q^{3} - 2 q^{5} + 2 q^{7} + q^{9} - 6 q^{13} + 2 q^{15} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrr} 1 & 2 & 5 & 10 \\ 2 & 1 & 10 & 5 \\ 5 & 10 & 1 & 2 \\ 10 & 5 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.