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Properties

Label 443822.k
Number of curves $3$
Conductor $443822$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("k1")
 
E.isogeny_class()
 

Elliptic curves in class 443822.k

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
443822.k1 443822k3 \([1, 1, 1, -14653207, -21595807443]\) \(15698803397448457/20709376\) \(459010088420245504\) \([]\) \(17971200\) \(2.6650\)  
443822.k2 443822k2 \([1, 1, 1, -228992, -12743583]\) \(59914169497/31554496\) \(699385244587585984\) \([]\) \(5990400\) \(2.1157\)  
443822.k3 443822k1 \([1, 1, 1, -130677, 18127327]\) \(11134383337/316\) \(7003938116764\) \([]\) \(1996800\) \(1.5664\) \(\Gamma_0(N)\)-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 0 curves highlighted, and conditionally curve 443822.k1.

Rank

sage: E.rank()
 

The elliptic curves in class 443822.k have rank \(0\).

Complex multiplication

The elliptic curves in class 443822.k do not have complex multiplication.

Modular form 443822.2.a.k

sage: E.q_eigenform(10)
 
\(q + q^{2} - q^{3} + q^{4} - 3 q^{5} - q^{6} - q^{7} + q^{8} - 2 q^{9} - 3 q^{10} - q^{12} + 5 q^{13} - q^{14} + 3 q^{15} + q^{16} - 2 q^{18} - 2 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrr} 1 & 3 & 9 \\ 3 & 1 & 3 \\ 9 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.