Properties

Label 53312o
Number of curves $4$
Conductor $53312$
CM no
Rank $2$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("53312.ba1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 53312o

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
53312.ba3 53312o1 [0, 0, 0, -2156, -16464] [2] 36864 \(\Gamma_0(N)\)-optimal
53312.ba2 53312o2 [0, 0, 0, -17836, 905520] [2, 2] 73728  
53312.ba4 53312o3 [0, 0, 0, -2156, 2442160] [2] 147456  
53312.ba1 53312o4 [0, 0, 0, -284396, 58375856] [2] 147456  

Rank

sage: E.rank()
 

The elliptic curves in class 53312o have rank \(2\).

Modular form 53312.2.a.ba

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

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.