↑

Properties

Label 22050.ba
Number of curves $6$
Conductor $22050$
CM no
Rank $1$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
Copy content sage:E = EllipticCurve("ba1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 22050.ba have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(3\)\(1\)
\(5\)\(1\)
\(7\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(11\) \( 1 + 11 T^{2}\) 1.11.a
\(13\) \( 1 + 4 T + 13 T^{2}\) 1.13.e
\(17\) \( 1 + 6 T + 17 T^{2}\) 1.17.g
\(19\) \( 1 + 2 T + 19 T^{2}\) 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 22050.ba do not have complex multiplication.

Modular form 22050.2.a.ba

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} + q^{4} - q^{8} - 4 q^{13} + q^{16} - 6 q^{17} - 2 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 22050.ba

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
22050.ba1 22050bi6 \([1, -1, 0, -30103992, -63567163584]\) \(2251439055699625/25088\) \(33620319432000000\) \([2]\) \(995328\) \(2.7401\)  
22050.ba2 22050bi5 \([1, -1, 0, -1879992, -994555584]\) \(-548347731625/1835008\) \(-2459086221312000000\) \([2]\) \(497664\) \(2.3935\)  
22050.ba3 22050bi4 \([1, -1, 0, -391617, -77220459]\) \(4956477625/941192\) \(1261287296191125000\) \([2]\) \(331776\) \(2.1908\)  
22050.ba4 22050bi2 \([1, -1, 0, -115992, 15224166]\) \(128787625/98\) \(131329372781250\) \([2]\) \(110592\) \(1.6415\)  
22050.ba5 22050bi1 \([1, -1, 0, -5742, 340416]\) \(-15625/28\) \(-37522677937500\) \([2]\) \(55296\) \(1.2949\) \(\Gamma_0(N)\)-optimal
22050.ba6 22050bi3 \([1, -1, 0, 49383, -7101459]\) \(9938375/21952\) \(-29417779503000000\) \([2]\) \(165888\) \(1.8442\)