Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, 0, 0, -2168715, 1229095692]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 0, 0, -2168715, 1229095692]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 0, 0, -2168715, 1229095692]) 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 5415.e have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(3\)\(1 - T\)
\(5\)\(1 - T\)
\(19\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(2\) \( 1 + T + 2 T^{2}\) 1.2.b
\(7\) \( 1 - 4 T + 7 T^{2}\) 1.7.ae
\(11\) \( 1 - 4 T + 11 T^{2}\) 1.11.ae
\(13\) \( 1 + 2 T + 13 T^{2}\) 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 5415.e do not have complex multiplication.

Modular form 5415.2.a.e

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);
 
\(q - q^{2} + q^{3} - q^{4} + q^{5} - q^{6} + 4 q^{7} + 3 q^{8} + q^{9} - q^{10} + 4 q^{11} - q^{12} - 2 q^{13} - 4 q^{14} + q^{15} - q^{16} + 2 q^{17} - q^{18} + O(q^{20})\) 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 5415.e

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
5415.e1 5415l4 \([1, 0, 0, -2168715, 1229095692]\) \(23977812996389881/146611125\) \(6897449540026125\) \([2]\) \(103680\) \(2.2264\)  
5415.e2 5415l3 \([1, 0, 0, -446745, -93110850]\) \(209595169258201/41748046875\) \(1964073645263671875\) \([2]\) \(103680\) \(2.2264\)  
5415.e3 5415l2 \([1, 0, 0, -138090, 18437067]\) \(6189976379881/456890625\) \(21494821973765625\) \([2, 2]\) \(51840\) \(1.8798\)  
5415.e4 5415l1 \([1, 0, 0, 8115, 1272600]\) \(1256216039/15582375\) \(-733086559947375\) \([4]\) \(25920\) \(1.5332\) \(\Gamma_0(N)\)-optimal