Properties

Label 45414o
Number of curves $3$
Conductor $45414$
CM no
Rank $2$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
E = EllipticCurve("o1")
 
E.isogeny_class()
 

Elliptic curves in class 45414o

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
45414.a3 45414o1 \([1, -1, 0, 1104, -10184]\) \(9261/8\) \(-128481837336\) \([]\) \(48384\) \(0.81914\) \(\Gamma_0(N)\)-optimal
45414.a1 45414o2 \([1, -1, 0, -24126, -1455022]\) \(-132651/2\) \(-23415814854486\) \([]\) \(145152\) \(1.3684\)  
45414.a2 45414o3 \([1, -1, 0, -11511, 633181]\) \(-1167051/512\) \(-74005538305536\) \([]\) \(145152\) \(1.3684\)  

Rank

sage: E.rank()
 

The elliptic curves in class 45414o have rank \(2\).

Complex multiplication

The elliptic curves in class 45414o do not have complex multiplication.

Modular form 45414.2.a.o

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{4} - 3 q^{5} - q^{7} - q^{8} + 3 q^{10} - 3 q^{11} - 4 q^{13} + q^{14} + q^{16} - 2 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrr} 1 & 3 & 3 \\ 3 & 1 & 9 \\ 3 & 9 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.