Properties

Label 448.d
Number of curves $4$
Conductor $448$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("d1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 448.d

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
448.d1 448a4 [0, 0, 0, -1196, 15920] [4] 128  
448.d2 448a3 [0, 0, 0, -236, -1104] [2] 128  
448.d3 448a2 [0, 0, 0, -76, 240] [2, 2] 64  
448.d4 448a1 [0, 0, 0, 4, 16] [2] 32 \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 448.2.a.d

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 LMFDB numbering.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.