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Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, 1, 0, -546801, 155401866]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 1, 0, -546801, 155401866]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 1, 0, -546801, 155401866]) 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 2541.h have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(3\)\(1 + T\)
\(7\)\(1 + T\)
\(11\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(2\) \( 1 - T + 2 T^{2}\) 1.2.ab
\(5\) \( 1 + 2 T + 5 T^{2}\) 1.5.c
\(13\) \( 1 + 6 T + 13 T^{2}\) 1.13.g
\(17\) \( 1 + 2 T + 17 T^{2}\) 1.17.c
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 - 2 T + 29 T^{2}\) 1.29.ac
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 2541.h do not have complex multiplication.

Modular form 2541.2.a.h

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

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 4 & 8 \\ 2 & 1 & 2 & 4 & 2 & 4 \\ 4 & 2 & 1 & 2 & 4 & 2 \\ 8 & 4 & 2 & 1 & 8 & 4 \\ 4 & 2 & 4 & 8 & 1 & 8 \\ 8 & 4 & 2 & 4 & 8 & 1 \end{array}\right)\)

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 2541.h

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
2541.h1 2541b5 \([1, 1, 0, -546801, 155401866]\) \(10206027697760497/5557167\) \(9844860327687\) \([2]\) \(19200\) \(1.8217\)  
2541.h2 2541b3 \([1, 1, 0, -34366, 2388775]\) \(2533811507137/58110129\) \(102945638241369\) \([2, 2]\) \(9600\) \(1.4752\)  
2541.h3 2541b2 \([1, 1, 0, -4721, -71760]\) \(6570725617/2614689\) \(4632081059529\) \([2, 2]\) \(4800\) \(1.1286\)  
2541.h4 2541b1 \([1, 1, 0, -4116, -103341]\) \(4354703137/1617\) \(2864614137\) \([2]\) \(2400\) \(0.78202\) \(\Gamma_0(N)\)-optimal
2541.h5 2541b6 \([1, 1, 0, 3749, 7442824]\) \(3288008303/13504609503\) \(-23924239515744183\) \([2]\) \(19200\) \(1.8217\)  
2541.h6 2541b4 \([1, 1, 0, 15244, -499011]\) \(221115865823/190238433\) \(-337018988603913\) \([2]\) \(9600\) \(1.4752\)