Properties

Label 432450fz
Number of curves $6$
Conductor $432450$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("432450.fz1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 432450fz

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
432450.fz6 432450fz1 [1, -1, 1, 12968995, -101854831003] [2] 94371840 \(\Gamma_0(N)\)-optimal*
432450.fz5 432450fz2 [1, -1, 1, -263799005, -1558208047003] [2, 2] 188743680 \(\Gamma_0(N)\)-optimal*
432450.fz4 432450fz3 [1, -1, 1, -800037005, 6791017612997] [2] 377487360 \(\Gamma_0(N)\)-optimal*
432450.fz2 432450fz4 [1, -1, 1, -4155849005, -103117360747003] [2, 2] 377487360  
432450.fz3 432450fz5 [1, -1, 1, -4090981505, -106492157302003] [2] 754974720  
432450.fz1 432450fz6 [1, -1, 1, -66493516505, -6599575041592003] [2] 754974720  
*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 432450fz1.

Rank

sage: E.rank()
 

The elliptic curves in class 432450fz have rank \(0\).

Modular form 432450.2.a.fz

sage: E.q_eigenform(10)
 
\( q + q^{2} + q^{4} + q^{8} - 4q^{11} + 6q^{13} + q^{16} - 2q^{17} + 4q^{19} + O(q^{20}) \)

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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.