Properties

Label 198550p
Number of curves $2$
Conductor $198550$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("p1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 198550p have rank \(0\).

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
\(7\) \( 1 + 3 T + 7 T^{2}\) 1.7.d
\(13\) \( 1 - 2 T + 13 T^{2}\) 1.13.ac
\(17\) \( 1 + 4 T + 17 T^{2}\) 1.17.e
\(23\) \( 1 + 2 T + 23 T^{2}\) 1.23.c
\(29\) \( 1 + 29 T^{2}\) 1.29.a
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

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

Modular form 198550.2.a.p

Copy content sage:E.q_eigenform(10)
 
\(q + q^{2} - q^{3} + q^{4} - q^{6} - 3 q^{7} + q^{8} - 2 q^{9} + q^{11} - q^{12} - 6 q^{13} - 3 q^{14} + q^{16} + 7 q^{17} - 2 q^{18} + O(q^{20})\) 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 Cremona numbering.

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

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 198550p

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
198550.cd2 198550p1 \([1, 1, 1, 90062, 42032031]\) \(109902239/1100000\) \(-808601079687500000\) \([]\) \(3240000\) \(2.1173\) \(\Gamma_0(N)\)-optimal
198550.cd1 198550p2 \([1, 1, 1, -53608688, 151055552031]\) \(-23178622194826561/1610510\) \(-1183872840770468750\) \([]\) \(16200000\) \(2.9220\)