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Properties

Label 29624.n
Number of curves $2$
Conductor $29624$
CM no
Rank $1$
Graph

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Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([0, -1, 0, -1413664, -611787540]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, -1, 0, -1413664, -611787540]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, -1, 0, -1413664, -611787540]) 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 29624.n have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1\)
\(7\)\(1 + T\)
\(23\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(3\) \( 1 - 2 T + 3 T^{2}\) 1.3.ac
\(5\) \( 1 + 2 T + 5 T^{2}\) 1.5.c
\(11\) \( 1 + 2 T + 11 T^{2}\) 1.11.c
\(13\) \( 1 - 4 T + 13 T^{2}\) 1.13.ae
\(17\) \( 1 - 2 T + 17 T^{2}\) 1.17.ac
\(19\) \( 1 + 19 T^{2}\) 1.19.a
\(29\) \( 1 + 2 T + 29 T^{2}\) 1.29.c
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 29624.n do not have complex multiplication.

Modular form 29624.2.a.n

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);
 
\(q + 2 q^{3} - 2 q^{5} - q^{7} + q^{9} - 2 q^{11} + 4 q^{13} - 4 q^{15} + 2 q^{17} + O(q^{20})\) 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.

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

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 29624.n

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
29624.n1 29624l2 \([0, -1, 0, -1413664, -611787540]\) \(1030541881826/62236321\) \(18868652251800307712\) \([2]\) \(506880\) \(2.4511\)  
29624.n2 29624l1 \([0, -1, 0, -1392504, -632008036]\) \(1969910093092/7889\) \(1195883651400704\) \([2]\) \(253440\) \(2.1045\) \(\Gamma_0(N)\)-optimal