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Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([0, 0, 0, -33543003, 74744443402]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 0, 0, -33543003, 74744443402]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 0, 0, -33543003, 74744443402]) 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 251280.x have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1\)
\(5\)\(1 + T\)
\(349\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 - 4 T + 7 T^{2}\) 1.7.ae
\(11\) \( 1 + 4 T + 11 T^{2}\) 1.11.e
\(13\) \( 1 + 2 T + 13 T^{2}\) 1.13.c
\(17\) \( 1 - 6 T + 17 T^{2}\) 1.17.ag
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 + 6 T + 29 T^{2}\) 1.29.g
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 251280.x do not have complex multiplication.

Modular form 251280.2.a.x

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);
 
\(q - q^{5} + 4 q^{7} - 4 q^{11} - 2 q^{13} + 6 q^{17} - 4 q^{19} + O(q^{20})\) 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.

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

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 251280.x

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
251280.x1 251280x3 \([0, 0, 0, -33543003, 74744443402]\) \(1397790417785316543001/640892891563200\) \(1913695919921450188800\) \([4]\) \(23887872\) \(3.0406\)  
251280.x2 251280x4 \([0, 0, 0, -18520923, -30152197622]\) \(235301185027835613721/4636822725000000\) \(13845478467686400000000\) \([2]\) \(23887872\) \(3.0406\)  
251280.x3 251280x2 \([0, 0, 0, -2439003, 760469002]\) \(537369779909439001/227309898240000\) \(678743719186268160000\) \([2, 2]\) \(11943936\) \(2.6940\)  
251280.x4 251280x1 \([0, 0, 0, 510117, 87479818]\) \(4916382075769319/3952292659200\) \(-11801482643688652800\) \([2]\) \(5971968\) \(2.3474\) \(\Gamma_0(N)\)-optimal