Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, 0, 0, -52, 144]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 0, 0, -52, 144]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 0, 0, -52, 144]) 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 242.b have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(11\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(3\) \( 1 + 2 T + 3 T^{2}\) 1.3.c
\(5\) \( 1 + 3 T + 5 T^{2}\) 1.5.d
\(7\) \( 1 + 2 T + 7 T^{2}\) 1.7.c
\(13\) \( 1 + 5 T + 13 T^{2}\) 1.13.f
\(17\) \( 1 + 3 T + 17 T^{2}\) 1.17.d
\(19\) \( 1 + 2 T + 19 T^{2}\) 1.19.c
\(23\) \( 1 - 6 T + 23 T^{2}\) 1.23.ag
\(29\) \( 1 - 3 T + 29 T^{2}\) 1.29.ad
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 242.b do not have complex multiplication.

Modular form 242.2.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);
 
\(q + q^{2} - 2 q^{3} + q^{4} - 3 q^{5} - 2 q^{6} - 2 q^{7} + q^{8} + q^{9} - 3 q^{10} - 2 q^{12} - 5 q^{13} - 2 q^{14} + 6 q^{15} + q^{16} - 3 q^{17} + q^{18} - 2 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)

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 242.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
242.b1 242a2 \([1, 0, 0, -52, 144]\) \(-128667913/4096\) \(-495616\) \([]\) \(48\) \(-0.13121\)  
242.b2 242a1 \([1, 0, 0, 3, 1]\) \(24167/16\) \(-1936\) \([]\) \(16\) \(-0.68051\) \(\Gamma_0(N)\)-optimal