Properties

Label 433200.fl
Number of curves $4$
Conductor $433200$
CM no
Rank $1$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("433200.fl1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 433200.fl

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
433200.fl1 433200fl3 [0, -1, 0, -438979008, 3540231760512] [4] 79626240 \(\Gamma_0(N)\)-optimal*
433200.fl2 433200fl4 [0, -1, 0, -31771008, 36683440512] [2] 79626240  
433200.fl3 433200fl2 [0, -1, 0, -27439008, 55311040512] [2, 2] 39813120 \(\Gamma_0(N)\)-optimal*
433200.fl4 433200fl1 [0, -1, 0, -1447008, 1143712512] [2] 19906560 \(\Gamma_0(N)\)-optimal*
*optimality has not been proved rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 3 curves highlighted, and conditionally curve 433200.fl4.

Rank

sage: E.rank()
 

The elliptic curves in class 433200.fl have rank \(1\).

Modular form 433200.2.a.fl

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

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.