Properties

Label 20286x
Number of curves $4$
Conductor $20286$
CM no
Rank $1$
Graph

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Show commands: SageMath
E = EllipticCurve("x1")
 
E.isogeny_class()
 

Elliptic curves in class 20286x

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
20286.g4 20286x1 \([1, -1, 0, -264168, -85182224]\) \(-23771111713777/22848457968\) \(-1959623610746902128\) \([2]\) \(368640\) \(2.2061\) \(\Gamma_0(N)\)-optimal
20286.g3 20286x2 \([1, -1, 0, -4929948, -4210664900]\) \(154502321244119857/55101928644\) \(4725878679414669924\) \([2, 2]\) \(737280\) \(2.5527\)  
20286.g1 20286x3 \([1, -1, 0, -78872418, -269590189730]\) \(632678989847546725777/80515134\) \(6905470724975214\) \([2]\) \(1474560\) \(2.8993\)  
20286.g2 20286x4 \([1, -1, 0, -5639958, -2918020694]\) \(231331938231569617/90942310746882\) \(7799769227536681984722\) \([2]\) \(1474560\) \(2.8993\)  

Rank

sage: E.rank()
 

The elliptic curves in class 20286x have rank \(1\).

Complex multiplication

The elliptic curves in class 20286x do not have complex multiplication.

Modular form 20286.2.a.x

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