Properties

Label 663.a
Number of curves $6$
Conductor $663$
CM no
Rank $1$
Graph

Related objects

Downloads

Learn more about

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

Elliptic curves in class 663.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
663.a1 663b5 \([1, 1, 1, -20174, -1111138]\) \(908031902324522977/161726530797\) \(161726530797\) \([2]\) \(1024\) \(1.1552\)  
663.a2 663b3 \([1, 1, 1, -1389, -14094]\) \(296380748763217/92608836489\) \(92608836489\) \([2, 2]\) \(512\) \(0.80858\)  
663.a3 663b2 \([1, 1, 1, -544, 4496]\) \(17806161424897/668584449\) \(668584449\) \([2, 4]\) \(256\) \(0.46201\)  
663.a4 663b1 \([1, 1, 1, -539, 4592]\) \(17319700013617/25857\) \(25857\) \([4]\) \(128\) \(0.11544\) \(\Gamma_0(N)\)-optimal
663.a5 663b4 \([1, 1, 1, 221, 17042]\) \(1193377118543/124806800313\) \(-124806800313\) \([4]\) \(512\) \(0.80858\)  
663.a6 663b6 \([1, 1, 1, 3876, -89910]\) \(6439735268725823/7345472585373\) \(-7345472585373\) \([2]\) \(1024\) \(1.1552\)  

Rank

sage: E.rank()
 

The elliptic curves in class 663.a have rank \(1\).

Complex multiplication

The elliptic curves in class 663.a do not have complex multiplication.

Modular form 663.2.a.a

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

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 8 & 4 \\ 2 & 1 & 2 & 4 & 4 & 2 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 4 & 2 & 1 & 4 & 8 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.