Properties

Label 63.a
Number of curves $6$
Conductor $63$
CM no
Rank $0$
Graph

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

Elliptic curves in class 63.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
63.a1 63a5 \([1, -1, 0, -7056, 229905]\) \(53297461115137/147\) \(107163\) \([4]\) \(32\) \(0.62351\)  
63.a2 63a4 \([1, -1, 0, -441, 3672]\) \(13027640977/21609\) \(15752961\) \([2, 2]\) \(16\) \(0.27694\)  
63.a3 63a3 \([1, -1, 0, -351, -2430]\) \(6570725617/45927\) \(33480783\) \([2]\) \(16\) \(0.27694\)  
63.a4 63a6 \([1, -1, 0, -306, 5859]\) \(-4354703137/17294403\) \(-12607619787\) \([2]\) \(32\) \(0.62351\)  
63.a5 63a2 \([1, -1, 0, -36, 27]\) \(7189057/3969\) \(2893401\) \([2, 2]\) \(8\) \(-0.069636\)  
63.a6 63a1 \([1, -1, 0, 9, 0]\) \(103823/63\) \(-45927\) \([2]\) \(4\) \(-0.41621\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 63.a have rank \(0\).

Complex multiplication

The elliptic curves in class 63.a do not have complex multiplication.

Modular form 63.2.a.a

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