Properties

Label 32448bh
Number of curves $6$
Conductor $32448$
CM no
Rank $0$
Graph

Related objects

Downloads

Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("bh1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 32448bh

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
32448.ch5 32448bh1 \([0, 1, 0, 451, 4707]\) \(2048/3\) \(-14827957248\) \([2]\) \(18432\) \(0.63712\) \(\Gamma_0(N)\)-optimal
32448.ch4 32448bh2 \([0, 1, 0, -2929, 44591]\) \(35152/9\) \(711741947904\) \([2, 2]\) \(36864\) \(0.98370\)  
32448.ch3 32448bh3 \([0, 1, 0, -16449, -780129]\) \(1556068/81\) \(25622710124544\) \([2, 2]\) \(73728\) \(1.3303\)  
32448.ch2 32448bh4 \([0, 1, 0, -43489, 3475967]\) \(28756228/3\) \(948989263872\) \([2]\) \(73728\) \(1.3303\)  
32448.ch6 32448bh5 \([0, 1, 0, 10591, -3067713]\) \(207646/6561\) \(-4150879040176128\) \([2]\) \(147456\) \(1.6768\)  
32448.ch1 32448bh6 \([0, 1, 0, -259809, -51058305]\) \(3065617154/9\) \(5693935583232\) \([2]\) \(147456\) \(1.6768\)  

Rank

sage: E.rank()
 

The elliptic curves in class 32448bh have rank \(0\).

Complex multiplication

The elliptic curves in class 32448bh do not have complex multiplication.

Modular form 32448.2.a.bh

sage: E.q_eigenform(10)
 
\(q + q^{3} - 2q^{5} + q^{9} + 4q^{11} - 2q^{15} + 2q^{17} - 4q^{19} + O(q^{20})\)  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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.