Properties

Label 266805df
Number of curves $4$
Conductor $266805$
CM no
Rank $0$
Graph

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

Elliptic curves in class 266805df

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
266805.df3 266805df1 \([1, -1, 0, -10860075, -13762511064]\) \(932288503609/779625\) \(118456156298050349625\) \([2]\) \(13271040\) \(2.7797\) \(\Gamma_0(N)\)-optimal
266805.df2 266805df2 \([1, -1, 0, -13261320, -7225841925]\) \(1697509118089/833765625\) \(126682278263192735015625\) \([2, 2]\) \(26542080\) \(3.1262\)  
266805.df1 266805df3 \([1, -1, 0, -113313195, 459276030450]\) \(1058993490188089/13182390375\) \(2002931274193252602420375\) \([2]\) \(53084160\) \(3.4728\)  
266805.df4 266805df4 \([1, -1, 0, 48370635, -55409704344]\) \(82375335041831/56396484375\) \(-8568877047023318115234375\) \([2]\) \(53084160\) \(3.4728\)  

Rank

sage: E.rank()
 

The elliptic curves in class 266805df have rank \(0\).

Complex multiplication

The elliptic curves in class 266805df do not have complex multiplication.

Modular form 266805.2.a.df

sage: E.q_eigenform(10)
 
\(q + q^{2} - q^{4} - q^{5} - 3 q^{8} - q^{10} - 2 q^{13} - q^{16} - 6 q^{17} + 4 q^{19} + O(q^{20})\) 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 Cremona 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 Cremona labels.