Properties

Label 231.a
Number of curves $6$
Conductor $231$
CM no
Rank $0$
Graph

Related objects

Downloads

Learn more

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

Elliptic curves in class 231.a

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
231.a1 231a5 \([1, 1, 1, -4519, -118810]\) \(10206027697760497/5557167\) \(5557167\) \([2]\) \(160\) \(0.62279\)  
231.a2 231a3 \([1, 1, 1, -284, -1924]\) \(2533811507137/58110129\) \(58110129\) \([2, 2]\) \(80\) \(0.27622\)  
231.a3 231a2 \([1, 1, 1, -39, 36]\) \(6570725617/2614689\) \(2614689\) \([2, 4]\) \(40\) \(-0.070357\)  
231.a4 231a1 \([1, 1, 1, -34, 62]\) \(4354703137/1617\) \(1617\) \([4]\) \(20\) \(-0.41693\) \(\Gamma_0(N)\)-optimal
231.a5 231a6 \([1, 1, 1, 31, -5578]\) \(3288008303/13504609503\) \(-13504609503\) \([2]\) \(160\) \(0.62279\)  
231.a6 231a4 \([1, 1, 1, 126, 432]\) \(221115865823/190238433\) \(-190238433\) \([4]\) \(80\) \(0.27622\)  

Rank

sage: E.rank()
 

The elliptic curves in class 231.a have rank \(0\).

Complex multiplication

The elliptic curves in class 231.a do not have complex multiplication.

Modular form 231.2.a.a

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{3} - q^{4} - 2 q^{5} + q^{6} + q^{7} + 3 q^{8} + q^{9} + 2 q^{10} - q^{11} + q^{12} + 6 q^{13} - q^{14} + 2 q^{15} - q^{16} + 2 q^{17} - q^{18} + 4 q^{19} + O(q^{20})\)  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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.