Properties

Label 20800.dk
Number of curves $4$
Conductor $20800$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("dk1")
 
E.isogeny_class()
 

Elliptic curves in class 20800.dk

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
20800.dk1 20800de3 \([0, -1, 0, -332033, 70795937]\) \(988345570681/44994560\) \(184297717760000000\) \([2]\) \(331776\) \(2.0749\)  
20800.dk2 20800de1 \([0, -1, 0, -52033, -4524063]\) \(3803721481/26000\) \(106496000000000\) \([2]\) \(110592\) \(1.5256\) \(\Gamma_0(N)\)-optimal
20800.dk3 20800de2 \([0, -1, 0, -20033, -10060063]\) \(-217081801/10562500\) \(-43264000000000000\) \([2]\) \(221184\) \(1.8722\)  
20800.dk4 20800de4 \([0, -1, 0, 179967, 268939937]\) \(157376536199/7722894400\) \(-31632975462400000000\) \([2]\) \(663552\) \(2.4215\)  

Rank

sage: E.rank()
 

The elliptic curves in class 20800.dk have rank \(1\).

Complex multiplication

The elliptic curves in class 20800.dk do not have complex multiplication.

Modular form 20800.2.a.dk

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

\(\left(\begin{array}{rrrr} 1 & 3 & 6 & 2 \\ 3 & 1 & 2 & 6 \\ 6 & 2 & 1 & 3 \\ 2 & 6 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.