Properties

Label 20631a
Number of curves $4$
Conductor $20631$
CM no
Rank $1$
Graph

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

Elliptic curves in class 20631a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
20631.g4 20631a1 \([1, 1, 0, 254, 3415]\) \(12167/39\) \(-5773399671\) \([2]\) \(12672\) \(0.55623\) \(\Gamma_0(N)\)-optimal
20631.g3 20631a2 \([1, 1, 0, -2391, 37800]\) \(10218313/1521\) \(225162587169\) \([2, 2]\) \(25344\) \(0.90281\)  
20631.g2 20631a3 \([1, 1, 0, -10326, -370059]\) \(822656953/85683\) \(12684159077187\) \([2]\) \(50688\) \(1.2494\)  
20631.g1 20631a4 \([1, 1, 0, -36776, 2699199]\) \(37159393753/1053\) \(155881791117\) \([2]\) \(50688\) \(1.2494\)  

Rank

sage: E.rank()
 

The elliptic curves in class 20631a have rank \(1\).

Complex multiplication

The elliptic curves in class 20631a do not have complex multiplication.

Modular form 20631.2.a.a

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