Properties

Label 209814ce
Number of curves $6$
Conductor $209814$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("ce1")
 
E.isogeny_class()
 

Elliptic curves in class 209814ce

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
209814.bu5 209814ce1 \([1, 0, 1, -9931925, -9126414112]\) \(2533811507137/625016832\) \(26726454678117956517888\) \([2]\) \(17694720\) \(3.0138\) \(\Gamma_0(N)\)-optimal
209814.bu4 209814ce2 \([1, 0, 1, -54692245, 148089733856]\) \(423108074414017/23284318464\) \(995664836973480278558976\) \([2, 2]\) \(35389440\) \(3.3603\)  
209814.bu2 209814ce3 \([1, 0, 1, -863175525, 9760955933056]\) \(1663303207415737537/5483698704\) \(234489404728399659029136\) \([2, 2]\) \(70778880\) \(3.7069\)  
209814.bu6 209814ce4 \([1, 0, 1, 37625915, 597236045888]\) \(137763859017023/3683199928848\) \(-157497959941026602927129232\) \([2]\) \(70778880\) \(3.7069\)  
209814.bu1 209814ce5 \([1, 0, 1, -13810797465, 624705670448968]\) \(6812873765474836663297/74052\) \(3166550595910976868\) \([2]\) \(141557760\) \(4.0535\)  
209814.bu3 209814ce6 \([1, 0, 1, -851286065, 10042912099064]\) \(-1595514095015181697/95635786040388\) \(-4089498666836889000521941092\) \([2]\) \(141557760\) \(4.0535\)  

Rank

sage: E.rank()
 

The elliptic curves in class 209814ce have rank \(1\).

Complex multiplication

The elliptic curves in class 209814ce do not have complex multiplication.

Modular form 209814.2.a.ce

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{3} + q^{4} + 2 q^{5} - q^{6} - q^{8} + q^{9} - 2 q^{10} + q^{12} + 2 q^{13} + 2 q^{15} + q^{16} - q^{18} + 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.