Properties

Label 436800.fz
Number of curves $4$
Conductor $436800$
CM no
Rank $1$
Graph

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

Elliptic curves in class 436800.fz

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
436800.fz1 436800fz4 \([0, -1, 0, -667233, -209553663]\) \(8020417344913/187278\) \(767090688000000\) \([2]\) \(4718592\) \(1.9684\)  
436800.fz2 436800fz2 \([0, -1, 0, -43233, -3009663]\) \(2181825073/298116\) \(1221083136000000\) \([2, 2]\) \(2359296\) \(1.6218\)  
436800.fz3 436800fz1 \([0, -1, 0, -11233, 414337]\) \(38272753/4368\) \(17891328000000\) \([2]\) \(1179648\) \(1.2753\) \(\Gamma_0(N)\)-optimal*
436800.fz4 436800fz3 \([0, -1, 0, 68767, -16113663]\) \(8780064047/32388174\) \(-132661960704000000\) \([2]\) \(4718592\) \(1.9684\)  
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 0 curves highlighted, and conditionally curve 436800.fz1.

Rank

sage: E.rank()
 

The elliptic curves in class 436800.fz have rank \(1\).

Complex multiplication

The elliptic curves in class 436800.fz do not have complex multiplication.

Modular form 436800.2.a.fz

sage: E.q_eigenform(10)
 
\(q - q^{3} + q^{7} + q^{9} - 4 q^{11} - q^{13} - 6 q^{17} - 4 q^{19} + O(q^{20})\) 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.