Properties

Label 411840.gz
Number of curves $4$
Conductor $411840$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("gz1")
 
E.isogeny_class()
 

Elliptic curves in class 411840.gz

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
411840.gz1 411840gz3 \([0, 0, 0, -3735372, 2778523504]\) \(30161840495801041/2799263610\) \(534947606479503360\) \([2]\) \(12582912\) \(2.4399\) \(\Gamma_0(N)\)-optimal*
411840.gz2 411840gz4 \([0, 0, 0, -1373772, -589164176]\) \(1500376464746641/83599963590\) \(15976201835540643840\) \([2]\) \(12582912\) \(2.4399\)  
411840.gz3 411840gz2 \([0, 0, 0, -250572, 36682864]\) \(9104453457841/2226896100\) \(425566471952793600\) \([2, 2]\) \(6291456\) \(2.0933\) \(\Gamma_0(N)\)-optimal*
411840.gz4 411840gz1 \([0, 0, 0, 37428, 3620464]\) \(30342134159/47190000\) \(-9018149437440000\) \([2]\) \(3145728\) \(1.7468\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 3 curves highlighted, and conditionally curve 411840.gz1.

Rank

sage: E.rank()
 

The elliptic curves in class 411840.gz have rank \(1\).

Complex multiplication

The elliptic curves in class 411840.gz do not have complex multiplication.

Modular form 411840.2.a.gz

sage: E.q_eigenform(10)
 
\(q + q^{5} - 4 q^{7} - q^{11} - q^{13} - 6 q^{17} + 8 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.