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Properties

Label 198550.ce
Number of curves $3$
Conductor $198550$
CM no
Rank $1$
Graph

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Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, 1, 1, -273710388, -1743086990219]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 1, 1, -273710388, -1743086990219]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 1, 1, -273710388, -1743086990219]) 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 198550.ce have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(5\)\(1\)
\(11\)\(1 - T\)
\(19\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(3\) \( 1 + T + 3 T^{2}\) 1.3.b
\(7\) \( 1 + 3 T + 7 T^{2}\) 1.7.d
\(13\) \( 1 - 4 T + 13 T^{2}\) 1.13.ae
\(17\) \( 1 + 3 T + 17 T^{2}\) 1.17.d
\(23\) \( 1 + 4 T + 23 T^{2}\) 1.23.e
\(29\) \( 1 + 5 T + 29 T^{2}\) 1.29.f
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 198550.ce do not have complex multiplication.

Modular form 198550.2.a.ce

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);
 
\(q + q^{2} - q^{3} + q^{4} - q^{6} - 3 q^{7} + q^{8} - 2 q^{9} + q^{11} - q^{12} + 4 q^{13} - 3 q^{14} + q^{16} - 3 q^{17} - 2 q^{18} + O(q^{20})\) 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.

\(\left(\begin{array}{rrr} 1 & 25 & 5 \\ 25 & 1 & 5 \\ 5 & 5 & 1 \end{array}\right)\)

Isogeny graph

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

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

Elliptic curves in class 198550.ce

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
198550.ce1 198550j3 \([1, 1, 1, -273710388, -1743086990219]\) \(-24680042791780949/369098752\) \(-33915187429376000000000\) \([]\) \(40500000\) \(3.4601\)  
198550.ce2 198550j1 \([1, 1, 1, -252888, 48891031]\) \(-19465109/22\) \(-2021502699218750\) \([]\) \(1620000\) \(1.8506\) \(\Gamma_0(N)\)-optimal
198550.ce3 198550j2 \([1, 1, 1, 1777737, -512915219]\) \(6761990971/5153632\) \(-473549136308187500000\) \([]\) \(8100000\) \(2.6553\)