Properties

Label 21840k
Number of curves $4$
Conductor $21840$
CM no
Rank $0$
Graph

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

Elliptic curves in class 21840k

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
21840.bk3 21840k1 \([0, 1, 0, -1614796, -790352116]\) \(1819018058610682173904/4844385\) \(1240162560\) \([2]\) \(172032\) \(1.8659\) \(\Gamma_0(N)\)-optimal
21840.bk2 21840k2 \([0, 1, 0, -1614816, -790331580]\) \(454771411897393003396/23468066028225\) \(24031299612902400\) \([2, 2]\) \(344064\) \(2.2125\)  
21840.bk4 21840k3 \([0, 1, 0, -1526936, -880074636]\) \(-192245661431796830258/51935513760073125\) \(-106363932180629760000\) \([2]\) \(688128\) \(2.5590\)  
21840.bk1 21840k4 \([0, 1, 0, -1703016, -699273900]\) \(266716694084614489298/51372277695070605\) \(105210424719504599040\) \([2]\) \(688128\) \(2.5590\)  

Rank

sage: E.rank()
 

The elliptic curves in class 21840k have rank \(0\).

Complex multiplication

The elliptic curves in class 21840k do not have complex multiplication.

Modular form 21840.2.a.k

sage: E.q_eigenform(10)
 
\(q + q^{3} - q^{5} - q^{7} + q^{9} - q^{13} - q^{15} - 6 q^{17} - 8 q^{19} + O(q^{20})\) 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.