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Properties

Label 283920.w
Number of curves $4$
Conductor $283920$
CM no
Rank $0$
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, -1629216, -712825920]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([0, -1, 0, -1629216, -712825920]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([0, -1, 0, -1629216, -712825920]) 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 283920.w have rank \(0\).

L-function data

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

Complex multiplication

The elliptic curves in class 283920.w do not have complex multiplication.

Modular form 283920.2.a.w

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);
 
\(q - q^{3} - q^{5} - q^{7} + q^{9} + 4 q^{11} + q^{15} - 2 q^{17} + 4 q^{19} + O(q^{20})\) 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.

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

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 283920.w

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
283920.w1 283920w4 \([0, -1, 0, -1629216, -712825920]\) \(24190225473961/2879296875\) \(56925454622400000000\) \([2]\) \(8257536\) \(2.5217\)  
283920.w2 283920w2 \([0, -1, 0, -398896, 85405696]\) \(355045312441/46580625\) \(920927354780160000\) \([2, 2]\) \(4128768\) \(2.1752\)  
283920.w3 283920w1 \([0, -1, 0, -385376, 92208960]\) \(320153881321/6825\) \(134934410956800\) \([2]\) \(2064384\) \(1.8286\) \(\Gamma_0(N)\)-optimal
283920.w4 283920w3 \([0, -1, 0, 615104, 448012096]\) \(1301812981559/5143122075\) \(-101682658999126732800\) \([2]\) \(8257536\) \(2.5217\)