Properties

Label 4032bm
Number of curves $6$
Conductor $4032$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("bm1")
 
E.isogeny_class()
 

Elliptic curves in class 4032bm

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
4032.k6 4032bm1 \([0, 0, 0, 564, -1136]\) \(103823/63\) \(-12039487488\) \([2]\) \(2048\) \(0.62351\) \(\Gamma_0(N)\)-optimal
4032.k5 4032bm2 \([0, 0, 0, -2316, -9200]\) \(7189057/3969\) \(758487711744\) \([2, 2]\) \(4096\) \(0.97009\)  
4032.k2 4032bm3 \([0, 0, 0, -28236, -1823600]\) \(13027640977/21609\) \(4129544208384\) \([2, 2]\) \(8192\) \(1.3167\)  
4032.k3 4032bm4 \([0, 0, 0, -22476, 1289104]\) \(6570725617/45927\) \(8776786378752\) \([2]\) \(8192\) \(1.3167\)  
4032.k1 4032bm5 \([0, 0, 0, -451596, -116808176]\) \(53297461115137/147\) \(28092137472\) \([2]\) \(16384\) \(1.6632\)  
4032.k4 4032bm6 \([0, 0, 0, -19596, -2960624]\) \(-4354703137/17294403\) \(-3305011881443328\) \([2]\) \(16384\) \(1.6632\)  

Rank

sage: E.rank()
 

The elliptic curves in class 4032bm have rank \(0\).

Complex multiplication

The elliptic curves in class 4032bm do not have complex multiplication.

Modular form 4032.2.a.bm

sage: E.q_eigenform(10)
 
\(q - 2 q^{5} + q^{7} - 4 q^{11} + 2 q^{13} + 6 q^{17} + 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.