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Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([0, 1, 0, -12625928, 17263833076]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 1, 0, -12625928, 17263833076]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 1, 0, -12625928, 17263833076]) 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 32368.f have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(7\)\(1 - T\)
\(17\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(3\) \( 1 + 2 T + 3 T^{2}\) 1.3.c
\(5\) \( 1 + 5 T^{2}\) 1.5.a
\(11\) \( 1 + 11 T^{2}\) 1.11.a
\(13\) \( 1 + 4 T + 13 T^{2}\) 1.13.e
\(19\) \( 1 + 2 T + 19 T^{2}\) 1.19.c
\(23\) \( 1 + 23 T^{2}\) 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 32368.f do not have complex multiplication.

Modular form 32368.2.a.f

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

\(\left(\begin{array}{rrrrrr} 1 & 2 & 3 & 9 & 18 & 6 \\ 2 & 1 & 6 & 18 & 9 & 3 \\ 3 & 6 & 1 & 3 & 6 & 2 \\ 9 & 18 & 3 & 1 & 2 & 6 \\ 18 & 9 & 6 & 2 & 1 & 3 \\ 6 & 3 & 2 & 6 & 3 & 1 \end{array}\right)\)

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 32368.f

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
32368.f1 32368bd6 \([0, 1, 0, -12625928, 17263833076]\) \(2251439055699625/25088\) \(2480387404070912\) \([2]\) \(663552\) \(2.5229\)  
32368.f2 32368bd5 \([0, 1, 0, -788488, 270004212]\) \(-548347731625/1835008\) \(-181422621554900992\) \([2]\) \(331776\) \(2.1763\)  
32368.f3 32368bd4 \([0, 1, 0, -164248, 20946324]\) \(4956477625/941192\) \(93053283705847808\) \([2]\) \(221184\) \(1.9735\)  
32368.f4 32368bd2 \([0, 1, 0, -48648, -4143500]\) \(128787625/98\) \(9689013297152\) \([2]\) \(73728\) \(1.4242\)  
32368.f5 32368bd1 \([0, 1, 0, -2408, -92876]\) \(-15625/28\) \(-2768289513472\) \([2]\) \(36864\) \(1.0777\) \(\Gamma_0(N)\)-optimal
32368.f6 32368bd3 \([0, 1, 0, 20712, 1932436]\) \(9938375/21952\) \(-2170338978562048\) \([2]\) \(110592\) \(1.6270\)