Properties

Label 38640.f
Number of curves $4$
Conductor $38640$
CM no
Rank $1$
Graph

Related objects

Downloads

Learn more

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

Elliptic curves in class 38640.f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
38640.f1 38640bj4 \([0, -1, 0, -45536, -3195264]\) \(2549399737314529/388286718750\) \(1590422400000000\) \([2]\) \(196608\) \(1.6407\)  
38640.f2 38640bj2 \([0, -1, 0, -12416, 487680]\) \(51682540549249/5249002500\) \(21499914240000\) \([2, 2]\) \(98304\) \(1.2941\)  
38640.f3 38640bj1 \([0, -1, 0, -12096, 516096]\) \(47788676405569/579600\) \(2374041600\) \([2]\) \(49152\) \(0.94755\) \(\Gamma_0(N)\)-optimal
38640.f4 38640bj3 \([0, -1, 0, 15584, 2346880]\) \(102181603702751/642612880350\) \(-2632142357913600\) \([2]\) \(196608\) \(1.6407\)  

Rank

sage: E.rank()
 

The elliptic curves in class 38640.f have rank \(1\).

Complex multiplication

The elliptic curves in class 38640.f do not have complex multiplication.

Modular form 38640.2.a.f

sage: E.q_eigenform(10)
 
\(q - q^{3} - q^{5} - q^{7} + q^{9} - 6q^{13} + q^{15} - 2q^{17} - 4q^{19} + O(q^{20})\)  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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.