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Properties

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

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Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, 1, 1, -4519, -118810]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 1, 1, -4519, -118810]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 1, 1, -4519, -118810]) ellisomat(E)
 

Rank

Copy content comment:Mordell-Weil rank
 
Copy content sage:E.rank()
 
Copy content gp:[lower,upper] = ellrank(E)
 
Copy content magma:Rank(E);
 

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

L-function data

Bad L-factors:
Prime L-Factor
\(3\)\(1 + T\)
\(7\)\(1 - T\)
\(11\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(2\) \( 1 + T + 2 T^{2}\) 1.2.b
\(5\) \( 1 + 2 T + 5 T^{2}\) 1.5.c
\(13\) \( 1 - 6 T + 13 T^{2}\) 1.13.ag
\(17\) \( 1 - 2 T + 17 T^{2}\) 1.17.ac
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 + 2 T + 29 T^{2}\) 1.29.c
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

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

Modular form 231.2.a.a

Copy content comment:q-expansion of modular form
 
Copy content sage:E.q_eigenform(20)
 
Copy content gp:Ser(ellan(E,20),q)*q
 
Copy content magma:ModularForm(E);
 
\(q - q^{2} - q^{3} - q^{4} - 2 q^{5} + q^{6} + q^{7} + 3 q^{8} + q^{9} + 2 q^{10} - q^{11} + q^{12} + 6 q^{13} - q^{14} + 2 q^{15} - q^{16} + 2 q^{17} - q^{18} + 4 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content comment:Isogeny matrix
 
Copy content sage:E.isogeny_class().matrix()
 
Copy content gp:ellisomat(E)
 

The \((i,j)\)-th 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

Copy content comment:Isogeny graph
 
Copy content sage:E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with LMFDB labels, and the \( \Gamma_0(N) \)-optimal curve is highlighted in blue.

Elliptic curves in class 231.a

Copy content comment:List of curves in the isogeny class
 
Copy content sage:E.isogeny_class().curves
 
Copy content magma:IsogenousCurves(E);
 
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 \([1, 1, 1, 126, 432]\) \(221115865823/190238433\) \(-190238433\) \([4]\) \(80\) \(0.27622\)