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Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([0, 1, 0, -384481, -91889281]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 1, 0, -384481, -91889281]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 1, 0, -384481, -91889281]) 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 18240.ce have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 - T\)
\(5\)\(1 + T\)
\(19\)\(1 - T\)
 
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
\(17\) \( 1 - 2 T + 17 T^{2}\) 1.17.ac
\(23\) \( 1 + 4 T + 23 T^{2}\) 1.23.e
\(29\) \( 1 - 2 T + 29 T^{2}\) 1.29.ac
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 18240.ce do not have complex multiplication.

Modular form 18240.2.a.ce

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);
 
\(q + q^{3} - q^{5} + 4 q^{7} + q^{9} - 4 q^{11} - 2 q^{13} - q^{15} + 2 q^{17} + q^{19} + O(q^{20})\) 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 labeled with LMFDB labels, and the \( \Gamma_0(N) \)-optimal curve is highlighted in blue.

Elliptic curves in class 18240.ce

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
18240.ce1 18240bj3 \([0, 1, 0, -384481, -91889281]\) \(23977812996389881/146611125\) \(38433226752000\) \([2]\) \(147456\) \(1.7939\)  
18240.ce2 18240bj4 \([0, 1, 0, -79201, 6921215]\) \(209595169258201/41748046875\) \(10944000000000000\) \([2]\) \(147456\) \(1.7939\)  
18240.ce3 18240bj2 \([0, 1, 0, -24481, -1385281]\) \(6189976379881/456890625\) \(119771136000000\) \([2, 2]\) \(73728\) \(1.4473\)  
18240.ce4 18240bj1 \([0, 1, 0, 1439, -94465]\) \(1256216039/15582375\) \(-4084826112000\) \([2]\) \(36864\) \(1.1007\) \(\Gamma_0(N)\)-optimal