↑

Properties

Label 81225.d
Number of curves $2$
Conductor $81225$
CM no
Rank $1$
Graph

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([0, 0, 1, -676875, -219702344]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 0, 1, -676875, -219702344]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 0, 1, -676875, -219702344]) ellisomat(E)
 

Rank

Copy content comment:Mordell-Weil rank
 
Copy content sage:E.rank()
 
Copy content gp:[lower,upper] = ellrank(E)
 
Copy content magma:Rank(E);
 

The elliptic curves in class 81225.d have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(3\)\(1\)
\(5\)\(1\)
\(19\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(2\) \( 1 + 2 T + 2 T^{2}\) 1.2.c
\(7\) \( 1 - 3 T + 7 T^{2}\) 1.7.ad
\(11\) \( 1 + 2 T + 11 T^{2}\) 1.11.c
\(13\) \( 1 - T + 13 T^{2}\) 1.13.ab
\(17\) \( 1 - 2 T + 17 T^{2}\) 1.17.ac
\(23\) \( 1 - 6 T + 23 T^{2}\) 1.23.ag
\(29\) \( 1 - 10 T + 29 T^{2}\) 1.29.ak
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 81225.d do not have complex multiplication.

Modular form 81225.2.a.d

Copy content comment:q-expansion of modular form
 
Copy content sage:E.q_eigenform(20)
 
Copy content gp:Ser(ellan(E,20),q)*q
 
Copy content magma:ModularForm(E);
 
\(q - 2 q^{2} + 2 q^{4} + 3 q^{7} - 2 q^{11} + q^{13} - 6 q^{14} - 4 q^{16} + 2 q^{17} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content comment:Isogeny matrix
 
Copy content sage:E.isogeny_class().matrix()
 
Copy content gp:ellisomat(E)
 

The \((i,j)\)-th 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels, and the \( \Gamma_0(N) \)-optimal curve is highlighted in blue.

Elliptic curves in class 81225.d

Copy content comment:List of curves in the isogeny class
 
Copy content sage:E.isogeny_class().curves
 
Copy content magma:IsogenousCurves(E);
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
81225.d1 81225bl2 \([0, 0, 1, -676875, -219702344]\) \(-102400/3\) \(-1004778727998046875\) \([]\) \(1620000\) \(2.2331\)  
81225.d2 81225bl1 \([0, 0, 1, 5415, 677326]\) \(20480/243\) \(-208350917037675\) \([]\) \(324000\) \(1.4284\) \(\Gamma_0(N)\)-optimal