↑

Properties

Label 304950.p
Number of curves $3$
Conductor $304950$
CM no
Rank $1$
Graph

Related objects

Downloads

Learn more

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

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 304950.p have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(3\)\(1 + T\)
\(5\)\(1\)
\(19\)\(1 - T\)
\(107\)\(1 - T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 - T + 7 T^{2}\) 1.7.ab
\(11\) \( 1 - 3 T + 11 T^{2}\) 1.11.ad
\(13\) \( 1 - 4 T + 13 T^{2}\) 1.13.ae
\(17\) \( 1 + 17 T^{2}\) 1.17.a
\(23\) \( 1 - 3 T + 23 T^{2}\) 1.23.ad
\(29\) \( 1 + 6 T + 29 T^{2}\) 1.29.g
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 304950.p do not have complex multiplication.

Modular form 304950.2.a.p

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} - q^{3} + q^{4} + q^{6} + q^{7} - q^{8} + q^{9} + 3 q^{11} - q^{12} + 4 q^{13} - q^{14} + q^{16} - q^{18} + q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrr} 1 & 9 & 3 \\ 9 & 1 & 3 \\ 3 & 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 304950.p

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
304950.p1 304950p3 \([1, 1, 0, -4140969750, -102633882249750]\) \(-502585390635922447924416132961/378299752139695192905390\) \(-5910933627182737389146718750\) \([]\) \(317447424\) \(4.2619\)  
304950.p2 304950p1 \([1, 1, 0, -5781000, 23473224000]\) \(-1367453954870275397761/14445609579000000000\) \(-225712649671875000000000\) \([]\) \(35271936\) \(3.1632\) \(\Gamma_0(N)\)-optimal
304950.p3 304950p2 \([1, 1, 0, 51594000, -604705401000]\) \(972078350597918586282239/10673270244457546419000\) \(-166769847569649162796875000\) \([]\) \(105815808\) \(3.7126\)