Properties

Label 193200.df
Number of curves $6$
Conductor $193200$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("df1")
 
E.isogeny_class()
 

Elliptic curves in class 193200.df

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
193200.df1 193200ef5 \([0, -1, 0, -31653608, 68555993712]\) \(54804145548726848737/637608031452\) \(40806914012928000000\) \([2]\) \(12582912\) \(2.9138\)  
193200.df2 193200ef4 \([0, -1, 0, -7085608, -7257062288]\) \(614716917569296417/19093020912\) \(1221953338368000000\) \([2]\) \(6291456\) \(2.5673\)  
193200.df3 193200ef3 \([0, -1, 0, -2029608, 1013273712]\) \(14447092394873377/1439452851984\) \(92124982526976000000\) \([2, 2]\) \(6291456\) \(2.5673\)  
193200.df4 193200ef2 \([0, -1, 0, -461608, -103142288]\) \(169967019783457/26337394944\) \(1685593276416000000\) \([2, 2]\) \(3145728\) \(2.2207\)  
193200.df5 193200ef1 \([0, -1, 0, 50392, -8934288]\) \(221115865823/664731648\) \(-42542825472000000\) \([2]\) \(1572864\) \(1.8741\) \(\Gamma_0(N)\)-optimal
193200.df6 193200ef6 \([0, -1, 0, 2506392, 4896089712]\) \(27207619911317663/177609314617308\) \(-11366996135507712000000\) \([2]\) \(12582912\) \(2.9138\)  

Rank

sage: E.rank()
 

The elliptic curves in class 193200.df have rank \(1\).

Complex multiplication

The elliptic curves in class 193200.df do not have complex multiplication.

Modular form 193200.2.a.df

sage: E.q_eigenform(10)
 
\(q - q^{3} + q^{7} + q^{9} + 4 q^{11} + 2 q^{13} + 6 q^{17} - 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.