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Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, 1, 1, -87552, -10007679]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 1, 1, -87552, -10007679]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 1, 1, -87552, -10007679]) 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 114.b have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(3\)\(1 + T\)
\(19\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 - 2 T + 5 T^{2}\) 1.5.ac
\(7\) \( 1 + 7 T^{2}\) 1.7.a
\(11\) \( 1 + 4 T + 11 T^{2}\) 1.11.e
\(13\) \( 1 - 2 T + 13 T^{2}\) 1.13.ac
\(17\) \( 1 + 6 T + 17 T^{2}\) 1.17.g
\(23\) \( 1 + 4 T + 23 T^{2}\) 1.23.e
\(29\) \( 1 + 2 T + 29 T^{2}\) 1.29.c
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 114.b do not have complex multiplication.

Modular form 114.2.a.b

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);
 
\(q + q^{2} - q^{3} + q^{4} + 2 q^{5} - q^{6} + q^{8} + q^{9} + 2 q^{10} - 4 q^{11} - q^{12} + 2 q^{13} - 2 q^{15} + q^{16} - 6 q^{17} + q^{18} - q^{19} + O(q^{20})\) 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 114.b

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
114.b1 114c3 \([1, 1, 1, -87552, -10007679]\) \(74220219816682217473/16416\) \(16416\) \([2]\) \(240\) \(1.1014\)  
114.b2 114c2 \([1, 1, 1, -5472, -158079]\) \(18120364883707393/269485056\) \(269485056\) \([2, 2]\) \(120\) \(0.75487\)  
114.b3 114c4 \([1, 1, 1, -5312, -167551]\) \(-16576888679672833/2216253521952\) \(-2216253521952\) \([2]\) \(240\) \(1.1014\)  
114.b4 114c1 \([1, 1, 1, -352, -2431]\) \(4824238966273/537919488\) \(537919488\) \([4]\) \(60\) \(0.40829\) \(\Gamma_0(N)\)-optimal