Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([0, 1, 0, -57687361, 168624083135]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 1, 0, -57687361, 168624083135]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 1, 0, -57687361, 168624083135]) 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 162240.ei have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 - T\)
\(5\)\(1 + T\)
\(13\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 + 4 T + 7 T^{2}\) 1.7.e
\(11\) \( 1 + 11 T^{2}\) 1.11.a
\(17\) \( 1 - 6 T + 17 T^{2}\) 1.17.ag
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 - 6 T + 29 T^{2}\) 1.29.ag
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 162240.ei do not have complex multiplication.

Modular form 162240.2.a.ei

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);
 
\(q + q^{3} - q^{5} - 4 q^{7} + q^{9} - q^{15} + 6 q^{17} + 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrrrrrr} 1 & 4 & 2 & 12 & 3 & 6 & 4 & 12 \\ 4 & 1 & 2 & 3 & 12 & 6 & 4 & 12 \\ 2 & 2 & 1 & 6 & 6 & 3 & 2 & 6 \\ 12 & 3 & 6 & 1 & 4 & 2 & 12 & 4 \\ 3 & 12 & 6 & 4 & 1 & 2 & 12 & 4 \\ 6 & 6 & 3 & 2 & 2 & 1 & 6 & 2 \\ 4 & 4 & 2 & 12 & 12 & 6 & 1 & 3 \\ 12 & 12 & 6 & 4 & 4 & 2 & 3 & 1 \end{array}\right)\)

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 162240.ei

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
162240.ei1 162240ba7 \([0, 1, 0, -57687361, 168624083135]\) \(16778985534208729/81000\) \(102490840498176000\) \([2]\) \(10616832\) \(2.8864\)  
162240.ei2 162240ba8 \([0, 1, 0, -4905281, 567497919]\) \(10316097499609/5859375000\) \(7413978624000000000000\) \([2]\) \(10616832\) \(2.8864\)  
162240.ei3 162240ba6 \([0, 1, 0, -3607361, 2630931135]\) \(4102915888729/9000000\) \(11387871166464000000\) \([2, 2]\) \(5308416\) \(2.5398\)  
162240.ei4 162240ba5 \([0, 1, 0, -3120641, -2122863105]\) \(2656166199049/33750\) \(42704516874240000\) \([2]\) \(3538944\) \(2.3371\)  
162240.ei5 162240ba4 \([0, 1, 0, -741121, 211272959]\) \(35578826569/5314410\) \(6724424045085327360\) \([2]\) \(3538944\) \(2.3371\)  
162240.ei6 162240ba2 \([0, 1, 0, -200321, -31329921]\) \(702595369/72900\) \(92241756448358400\) \([2, 2]\) \(1769472\) \(1.9905\)  
162240.ei7 162240ba3 \([0, 1, 0, -146241, 70394559]\) \(-273359449/1536000\) \(-1943530012409856000\) \([2]\) \(2654208\) \(2.1932\)  
162240.ei8 162240ba1 \([0, 1, 0, 15999, -2386305]\) \(357911/2160\) \(-2733089079951360\) \([2]\) \(884736\) \(1.6439\) \(\Gamma_0(N)\)-optimal