↑

Properties

Label 36822i
Number of curves $6$
Conductor $36822$
CM no
Rank $1$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
Copy content sage:E = EllipticCurve("i1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 36822i have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(3\)\(1 + T\)
\(17\)\(1 + T\)
\(19\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 - 3 T + 5 T^{2}\) 1.5.ad
\(7\) \( 1 + T + 7 T^{2}\) 1.7.b
\(11\) \( 1 - 6 T + 11 T^{2}\) 1.11.ag
\(13\) \( 1 - 4 T + 13 T^{2}\) 1.13.ae
\(23\) \( 1 - 6 T + 23 T^{2}\) 1.23.ag
\(29\) \( 1 - 6 T + 29 T^{2}\) 1.29.ag
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 36822i do not have complex multiplication.

Modular form 36822.2.a.i

Copy content sage:E.q_eigenform(10)
 
\(q - q^{2} - q^{3} + q^{4} - 2 q^{5} + q^{6} - q^{8} + q^{9} + 2 q^{10} - 4 q^{11} - q^{12} + 2 q^{13} + 2 q^{15} + q^{16} + q^{17} - q^{18} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.

Elliptic curves in class 36822i

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
36822.e5 36822i1 \([1, 1, 0, -12281, -490971]\) \(4354703137/352512\) \(16584237603072\) \([2]\) \(110592\) \(1.2796\) \(\Gamma_0(N)\)-optimal
36822.e4 36822i2 \([1, 1, 0, -41161, 2633845]\) \(163936758817/30338064\) \(1427280948714384\) \([2, 2]\) \(221184\) \(1.6262\)  
36822.e6 36822i3 \([1, 1, 0, 81579, 15472449]\) \(1276229915423/2927177028\) \(-137711622125221668\) \([2]\) \(442368\) \(1.9727\)  
36822.e2 36822i4 \([1, 1, 0, -625981, 190361065]\) \(576615941610337/27060804\) \(1273099364748324\) \([2, 2]\) \(442368\) \(1.9727\)  
36822.e3 36822i5 \([1, 1, 0, -593491, 211044199]\) \(-491411892194497/125563633938\) \(-5907251780174709378\) \([2]\) \(884736\) \(2.3193\)  
36822.e1 36822i6 \([1, 1, 0, -10015591, 12195916411]\) \(2361739090258884097/5202\) \(244732672962\) \([2]\) \(884736\) \(2.3193\)