Properties

Label 20a
Number of curves $4$
Conductor $20$
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 20a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
20.a4 20a1 \([0, 1, 0, 4, 4]\) \(21296/25\) \(-6400\) \([6]\) \(1\) \(-0.58338\) \(\Gamma_0(N)\)-optimal
20.a3 20a2 \([0, 1, 0, -1, 0]\) \(16384/5\) \(80\) \([6]\) \(2\) \(-0.92995\)  
20.a2 20a3 \([0, 1, 0, -36, -140]\) \(-20720464/15625\) \(-4000000\) \([2]\) \(3\) \(-0.034070\)  
20.a1 20a4 \([0, 1, 0, -41, -116]\) \(488095744/125\) \(2000\) \([2]\) \(6\) \(-0.38064\)  

Rank

sage: E.rank()
 

The elliptic curves in class 20a have rank \(0\).

Complex multiplication

The elliptic curves in class 20a do not have complex multiplication.

Modular form 20.2.a.a

sage: E.q_eigenform(10)
 
\(q - 2q^{3} - q^{5} + 2q^{7} + q^{9} + 2q^{13} + 2q^{15} - 6q^{17} - 4q^{19} + O(q^{20})\)  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 Cremona numbering.

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.