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Properties

Label 198550.s
Number of curves $4$
Conductor $198550$
CM no
Rank $2$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("s1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 198550.s have rank \(2\).

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 \(\mathbb{F}_p\)
\(3\) \( 1 + 3 T^{2}\) 1.3.a
\(7\) \( 1 + 7 T^{2}\) 1.7.a
\(13\) \( 1 - 2 T + 13 T^{2}\) 1.13.ac
\(17\) \( 1 - 2 T + 17 T^{2}\) 1.17.ac
\(23\) \( 1 - 4 T + 23 T^{2}\) 1.23.ae
\(29\) \( 1 + 6 T + 29 T^{2}\) 1.29.g
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

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

Modular form 198550.2.a.s

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} + q^{4} - q^{8} - 3 q^{9} - q^{11} + 2 q^{13} + q^{16} + 2 q^{17} + 3 q^{18} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content sage:E.isogeny_class().matrix()
 

The \(i,j\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 198550.s

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
198550.s1 198550cm4 \([1, -1, 0, -48934129292, -4166435191470384]\) \(17628594000102642361428441/248187500000000\) \(182440618604492187500000000\) \([2]\) \(318504960\) \(4.5931\)  
198550.s2 198550cm3 \([1, -1, 0, -4505137292, 2577802257616]\) \(13756443594716753103321/7957003087464992000\) \(5849128443273603207780500000000\) \([2]\) \(318504960\) \(4.5931\)  
198550.s3 198550cm2 \([1, -1, 0, -3061137292, -64976849742384]\) \(4315493878427398863321/16147293184000000\) \(11869744275102736000000000000\) \([2, 2]\) \(159252480\) \(4.2466\)  
198550.s4 198550cm1 \([1, -1, 0, -103825292, -1947659086384]\) \(-168380411424176601/2131914391552000\) \(-1567152980736606208000000000\) \([2]\) \(79626240\) \(3.9000\) \(\Gamma_0(N)\)-optimal