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Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([0, 0, 0, -33836, 2394576]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 0, 0, -33836, 2394576]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 0, 0, -33836, 2394576]) 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 7616.e have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(7\)\(1 + T\)
\(17\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(3\) \( 1 + 3 T^{2}\) 1.3.a
\(5\) \( 1 + 2 T + 5 T^{2}\) 1.5.c
\(11\) \( 1 + 11 T^{2}\) 1.11.a
\(13\) \( 1 - 2 T + 13 T^{2}\) 1.13.ac
\(19\) \( 1 - 4 T + 19 T^{2}\) 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 7616.e do not have complex multiplication.

Modular form 7616.2.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);
 
\(q - 2 q^{5} - q^{7} - 3 q^{9} + 2 q^{13} + q^{17} + 4 q^{19} + O(q^{20})\) 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.

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

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 7616.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
7616.e1 7616i3 \([0, 0, 0, -33836, 2394576]\) \(16342588257633/8185058\) \(2145663844352\) \([4]\) \(12288\) \(1.3188\)  
7616.e2 7616i2 \([0, 0, 0, -2476, 23760]\) \(6403769793/2775556\) \(727595352064\) \([2, 2]\) \(6144\) \(0.97228\)  
7616.e3 7616i1 \([0, 0, 0, -1196, -15664]\) \(721734273/13328\) \(3493855232\) \([2]\) \(3072\) \(0.62570\) \(\Gamma_0(N)\)-optimal
7616.e4 7616i4 \([0, 0, 0, 8404, 176080]\) \(250404380127/196003234\) \(-51381071773696\) \([2]\) \(12288\) \(1.3188\)