Properties

Label 21.a
Number of curves $6$
Conductor $21$
CM no
Rank $0$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("a1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 21.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
21.a1 21a5 \([1, 0, 0, -784, -8515]\) \(53297461115137/147\) \(147\) \([2]\) \(4\) \(0.074205\)  
21.a2 21a2 \([1, 0, 0, -49, -136]\) \(13027640977/21609\) \(21609\) \([2, 2]\) \(2\) \(-0.27237\)  
21.a3 21a3 \([1, 0, 0, -39, 90]\) \(6570725617/45927\) \(45927\) \([8]\) \(2\) \(-0.27237\)  
21.a4 21a6 \([1, 0, 0, -34, -217]\) \(-4354703137/17294403\) \(-17294403\) \([2]\) \(4\) \(0.074205\)  
21.a5 21a1 \([1, 0, 0, -4, -1]\) \(7189057/3969\) \(3969\) \([2, 4]\) \(1\) \(-0.61894\) \(\Gamma_0(N)\)-optimal
21.a6 21a4 \([1, 0, 0, 1, 0]\) \(103823/63\) \(-63\) \([4]\) \(2\) \(-0.96552\)  

Rank

sage: E.rank()
 

The elliptic curves in class 21.a have rank \(0\).

Complex multiplication

The elliptic curves in class 21.a do not have complex multiplication.

Modular form 21.2.a.a

sage: E.q_eigenform(10)
 
\(q - q^{2} + q^{3} - q^{4} - 2q^{5} - q^{6} - q^{7} + 3q^{8} + q^{9} + 2q^{10} + 4q^{11} - q^{12} - 2q^{13} + q^{14} - 2q^{15} - q^{16} - 6q^{17} - q^{18} + 4q^{19} + O(q^{20})\)  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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.