Properties

Label 448b
Number of curves $4$
Conductor $448$
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 448b

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
448.e4 448b1 [0, 0, 0, 4, -16] [2] 32 \(\Gamma_0(N)\)-optimal
448.e3 448b2 [0, 0, 0, -76, -240] [2, 2] 64  
448.e1 448b3 [0, 0, 0, -1196, -15920] [2] 128  
448.e2 448b4 [0, 0, 0, -236, 1104] [4] 128  

Rank

sage: E.rank()
 

The elliptic curves in class 448b have rank \(1\).

Complex multiplication

The elliptic curves in class 448b do not have complex multiplication.

Modular form 448.2.a.b

sage: E.q_eigenform(10)
 
\( q - 2q^{5} + q^{7} - 3q^{9} - 4q^{11} - 2q^{13} - 6q^{17} + 8q^{19} + O(q^{20}) \)

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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.