Properties

Label 13872b
Number of curves $6$
Conductor $13872$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("b1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 13872b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
13872.p5 13872b1 \([0, -1, 0, 193, -1338]\) \(2048/3\) \(-1158603312\) \([2]\) \(5120\) \(0.42468\) \(\Gamma_0(N)\)-optimal
13872.p4 13872b2 \([0, -1, 0, -1252, -12320]\) \(35152/9\) \(55612958976\) \([2, 2]\) \(10240\) \(0.77125\)  
13872.p2 13872b3 \([0, -1, 0, -18592, -969488]\) \(28756228/3\) \(74150611968\) \([2]\) \(20480\) \(1.1178\)  
13872.p3 13872b4 \([0, -1, 0, -7032, 218880]\) \(1556068/81\) \(2002066523136\) \([2, 2]\) \(20480\) \(1.1178\)  
13872.p1 13872b5 \([0, -1, 0, -111072, 14285088]\) \(3065617154/9\) \(444903671808\) \([2]\) \(40960\) \(1.4644\)  
13872.p6 13872b6 \([0, -1, 0, 4528, 856992]\) \(207646/6561\) \(-324334776748032\) \([2]\) \(40960\) \(1.4644\)  

Rank

sage: E.rank()
 

The elliptic curves in class 13872b have rank \(1\).

Complex multiplication

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

Modular form 13872.2.a.b

sage: E.q_eigenform(10)
 
\(q - q^{3} + 2q^{5} + q^{9} + 4q^{11} - 2q^{13} - 2q^{15} + 4q^{19} + O(q^{20})\)  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 Cremona numbering.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.