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Properties

Label 4730k
Number of curves $2$
Conductor $4730$
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([1, -1, 1, -5582262, 5077966149]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, -1, 1, -5582262, 5077966149]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, -1, 1, -5582262, 5077966149]) 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 4730k have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(5\)\(1 - T\)
\(11\)\(1 - T\)
\(43\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(3\) \( 1 - 2 T + 3 T^{2}\) 1.3.ac
\(7\) \( 1 + 7 T^{2}\) 1.7.a
\(13\) \( 1 - 6 T + 13 T^{2}\) 1.13.ag
\(17\) \( 1 - 2 T + 17 T^{2}\) 1.17.ac
\(19\) \( 1 - 2 T + 19 T^{2}\) 1.19.ac
\(23\) \( 1 + 6 T + 23 T^{2}\) 1.23.g
\(29\) \( 1 - 2 T + 29 T^{2}\) 1.29.ac
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 4730k do not have complex multiplication.

Modular form 4730.2.a.k

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);
 
\(q + q^{2} - 3 q^{3} + q^{4} + q^{5} - 3 q^{6} + q^{7} + q^{8} + 6 q^{9} + q^{10} + q^{11} - 3 q^{12} + q^{14} - 3 q^{15} + q^{16} - 3 q^{17} + 6 q^{18} - q^{19} + O(q^{20})\) 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 Cremona numbering.

\(\left(\begin{array}{rr} 1 & 7 \\ 7 & 1 \end{array}\right)\)

Isogeny graph

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

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

Elliptic curves in class 4730k

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
4730.d1 4730k1 \([1, -1, 1, -5582262, 5077966149]\) \(-19237750463016353596082481/360317791790000000\) \(-360317791790000000\) \([7]\) \(219520\) \(2.4932\) \(\Gamma_0(N)\)-optimal
4730.d2 4730k2 \([1, -1, 1, 40144188, -96077802411]\) \(7154705394529607961737582319/8127389307855235414199390\) \(-8127389307855235414199390\) \([]\) \(1536640\) \(3.4662\)