↑

Properties

Label 214080.cj
Number of curves $4$
Conductor $214080$
CM no
Rank $1$
Graph

Related objects

Downloads

Learn more

Show commands: SageMath
Copy content sage:E = EllipticCurve([0, 1, 0, -5388069985, 147896275627775]) E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 214080.cj have rank \(1\).

L-function data

 
Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1 - T\)
\(5\)\(1 - T\)
\(223\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 + 7 T^{2}\) 1.7.a
\(11\) \( 1 + 4 T + 11 T^{2}\) 1.11.e
\(13\) \( 1 - 2 T + 13 T^{2}\) 1.13.ac
\(17\) \( 1 + 2 T + 17 T^{2}\) 1.17.c
\(19\) \( 1 - 4 T + 19 T^{2}\) 1.19.ae
\(23\) \( 1 - 4 T + 23 T^{2}\) 1.23.ae
\(29\) \( 1 - 6 T + 29 T^{2}\) 1.29.ag
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 214080.cj do not have complex multiplication.

Modular form 214080.2.a.cj

Copy content sage:E.q_eigenform(10)
 
\(q + q^{3} + q^{5} + q^{9} - 4 q^{11} + 2 q^{13} + q^{15} - 2 q^{17} + 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.

Elliptic curves in class 214080.cj

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
214080.cj1 214080bl3 \([0, 1, 0, -5388069985, 147896275627775]\) \(65990812340278189764070806169/2142150634958771454912000\) \(561551936050632184276451328000\) \([4]\) \(265752576\) \(4.4821\)  
214080.cj2 214080bl2 \([0, 1, 0, -821029985, -5840338260225]\) \(233486000400975208694166169/78913673205682176000000\) \(20686745948830348345344000000\) \([2, 2]\) \(132876288\) \(4.1356\)  
214080.cj3 214080bl1 \([0, 1, 0, -737143905, -7702055588097]\) \(168982070711351853939176089/37703877214076928000\) \(9883845188406982213632000\) \([2]\) \(66438144\) \(3.7890\) \(\Gamma_0(N)\)-optimal
214080.cj4 214080bl4 \([0, 1, 0, 2403832735, -40425700987137]\) \(5859985279907178462243106151/6084442029900375000000000\) \(-1594999971486203904000000000000\) \([2]\) \(265752576\) \(4.4821\)