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Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([0, -1, 0, -2720, -53568]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, -1, 0, -2720, -53568]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, -1, 0, -2720, -53568]) 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 7440.o have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 + T\)
\(5\)\(1 - T\)
\(31\)\(1 - T\)
 
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.ae
\(13\) \( 1 - 2 T + 13 T^{2}\) 1.13.ac
\(17\) \( 1 + 6 T + 17 T^{2}\) 1.17.g
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 + 6 T + 29 T^{2}\) 1.29.g
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 7440.o do not have complex multiplication.

Modular form 7440.2.a.o

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);
 
\(q - q^{3} + q^{5} + 4 q^{7} + q^{9} + 4 q^{11} + 2 q^{13} - q^{15} - 6 q^{17} + 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

Copy content comment:Isogeny graph
 
Copy content sage:E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labeled with LMFDB labels, and the \( \Gamma_0(N) \)-optimal curve is highlighted in blue.

Elliptic curves in class 7440.o

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
7440.o1 7440r3 \([0, -1, 0, -2720, -53568]\) \(543538277281/1569375\) \(6428160000\) \([2]\) \(8192\) \(0.75293\)  
7440.o2 7440r2 \([0, -1, 0, -240, 0]\) \(374805361/216225\) \(885657600\) \([2, 2]\) \(4096\) \(0.40636\)  
7440.o3 7440r1 \([0, -1, 0, -160, 832]\) \(111284641/465\) \(1904640\) \([2]\) \(2048\) \(0.059784\) \(\Gamma_0(N)\)-optimal
7440.o4 7440r4 \([0, -1, 0, 960, -960]\) \(23862997439/13852815\) \(-56741130240\) \([4]\) \(8192\) \(0.75293\)