Properties

Label 53312ba
Number of curves 4
Conductor 53312
CM no
Rank 1
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("53312.k1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 53312ba

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
53312.k4 53312ba1 [0, 1, 0, -9473, -222881] [2] 110592 \(\Gamma_0(N)\)-optimal
53312.k3 53312ba2 [0, 1, 0, -134913, -19114145] [2] 221184  
53312.k2 53312ba3 [0, 1, 0, -323073, 70562911] [2] 331776  
53312.k1 53312ba4 [0, 1, 0, -354433, 56005599] [2] 663552  

Rank

sage: E.rank()
 

The elliptic curves in class 53312ba have rank \(1\).

Modular form 53312.2.a.k

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

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.

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.