Properties

Label 147.a
Number of curves $6$
Conductor $147$
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 147.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
147.a1 147a5 \([1, 1, 1, -38417, 2882228]\) \(53297461115137/147\) \(17294403\) \([2]\) \(192\) \(1.0472\)  
147.a2 147a4 \([1, 1, 1, -2402, 44246]\) \(13027640977/21609\) \(2542277241\) \([2, 2]\) \(96\) \(0.70059\)  
147.a3 147a3 \([1, 1, 1, -1912, -32782]\) \(6570725617/45927\) \(5403265623\) \([2]\) \(96\) \(0.70059\)  
147.a4 147a6 \([1, 1, 1, -1667, 72764]\) \(-4354703137/17294403\) \(-2034669218547\) \([2]\) \(192\) \(1.0472\)  
147.a5 147a2 \([1, 1, 1, -197, 146]\) \(7189057/3969\) \(466948881\) \([2, 2]\) \(48\) \(0.35401\)  
147.a6 147a1 \([1, 1, 1, 48, 48]\) \(103823/63\) \(-7411887\) \([4]\) \(24\) \(0.0074397\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 147.a have rank \(0\).

Complex multiplication

The elliptic curves in class 147.a do not have complex multiplication.

Modular form 147.2.a.a

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