Properties

Label 49725.e
Number of curves $4$
Conductor $49725$
CM no
Rank $1$
Graph

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

Elliptic curves in class 49725.e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
49725.e1 49725j4 \([1, -1, 1, -3511355, 2533439022]\) \(420339554066191969/244298925\) \(2782717442578125\) \([2]\) \(786432\) \(2.2882\)  
49725.e2 49725j2 \([1, -1, 1, -220730, 39145272]\) \(104413920565969/2472575625\) \(28164181728515625\) \([2, 2]\) \(393216\) \(1.9416\)  
49725.e3 49725j1 \([1, -1, 1, -30605, -1161228]\) \(278317173889/109245825\) \(1244378225390625\) \([2]\) \(196608\) \(1.5950\) \(\Gamma_0(N)\)-optimal
49725.e4 49725j3 \([1, -1, 1, 27895, 122186022]\) \(210751100351/566398828125\) \(-6451636651611328125\) \([2]\) \(786432\) \(2.2882\)  

Rank

sage: E.rank()
 

The elliptic curves in class 49725.e have rank \(1\).

Complex multiplication

The elliptic curves in class 49725.e do not have complex multiplication.

Modular form 49725.2.a.e

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{4} - 4 q^{7} + 3 q^{8} - q^{13} + 4 q^{14} - q^{16} + q^{17} + 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.