Properties

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

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Show commands: SageMath
sage: E = EllipticCurve("e1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 448.e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
448.e1 448b3 \([0, 0, 0, -1196, -15920]\) \(1443468546/7\) \(917504\) \([2]\) \(128\) \(0.34454\)  
448.e2 448b4 \([0, 0, 0, -236, 1104]\) \(11090466/2401\) \(314703872\) \([4]\) \(128\) \(0.34454\)  
448.e3 448b2 \([0, 0, 0, -76, -240]\) \(740772/49\) \(3211264\) \([2, 2]\) \(64\) \(-0.0020328\)  
448.e4 448b1 \([0, 0, 0, 4, -16]\) \(432/7\) \(-114688\) \([2]\) \(32\) \(-0.34861\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 448.2.a.e

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