↑

Properties

Label 69696.d
Number of curves $2$
Conductor $69696$
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, 0, -48972, 125840]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, 0, 0, -48972, 125840]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, 0, 0, -48972, 125840]) 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 69696.d have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(3\)\(1\)
\(11\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(5\) \( 1 + 4 T + 5 T^{2}\) 1.5.e
\(7\) \( 1 + 4 T + 7 T^{2}\) 1.7.e
\(13\) \( 1 - 6 T + 13 T^{2}\) 1.13.ag
\(17\) \( 1 - 6 T + 17 T^{2}\) 1.17.ag
\(19\) \( 1 - 6 T + 19 T^{2}\) 1.19.ag
\(23\) \( 1 - 2 T + 23 T^{2}\) 1.23.ac
\(29\) \( 1 + 2 T + 29 T^{2}\) 1.29.c
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 69696.d do not have complex multiplication.

Modular form 69696.2.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);
 
\(q - 4 q^{5} - 4 q^{7} + 6 q^{13} + 6 q^{17} + 6 q^{19} + O(q^{20})\) 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.

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

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 69696.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
69696.d1 69696fj2 \([0, 0, 0, -48972, 125840]\) \(102129622/59049\) \(7509794573647872\) \([2]\) \(737280\) \(1.7357\)  
69696.d2 69696fj1 \([0, 0, 0, -33132, -2313520]\) \(63253004/243\) \(15452252209152\) \([2]\) \(368640\) \(1.3892\) \(\Gamma_0(N)\)-optimal