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Properties

Label 166950.w
Number of curves $4$
Conductor $166950$
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([1, -1, 0, -156096792, 750683824366]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, -1, 0, -156096792, 750683824366]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, -1, 0, -156096792, 750683824366]) 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 166950.w have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(3\)\(1\)
\(5\)\(1\)
\(7\)\(1 + T\)
\(53\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(11\) \( 1 + 11 T^{2}\) 1.11.a
\(13\) \( 1 - 2 T + 13 T^{2}\) 1.13.ac
\(17\) \( 1 - 2 T + 17 T^{2}\) 1.17.ac
\(19\) \( 1 + 4 T + 19 T^{2}\) 1.19.e
\(23\) \( 1 + 8 T + 23 T^{2}\) 1.23.i
\(29\) \( 1 + 2 T + 29 T^{2}\) 1.29.c
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 166950.w do not have complex multiplication.

Modular form 166950.2.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);
 
\(q - q^{2} + q^{4} - q^{7} - q^{8} + 2 q^{13} + q^{14} + q^{16} + 2 q^{17} - 4 q^{19} + O(q^{20})\) 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.

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

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 166950.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
166950.w1 166950ey3 \([1, -1, 0, -156096792, 750683824366]\) \(36928196050908253259449/452758954469850\) \(5157207465758135156250\) \([2]\) \(23592960\) \(3.3142\)  
166950.w2 166950ey4 \([1, -1, 0, -36806292, -73825431134]\) \(484108118865316036729/73399966614843750\) \(836071494722204589843750\) \([2]\) \(23592960\) \(3.3142\)  
166950.w3 166950ey2 \([1, -1, 0, -10015542, 11074455616]\) \(9754377335041367449/995626517602500\) \(11340808302065976562500\) \([2, 2]\) \(11796480\) \(2.9676\)  
166950.w4 166950ey1 \([1, -1, 0, 788958, 842594116]\) \(4768013769464231/29697948831600\) \(-338278198409943750000\) \([2]\) \(5898240\) \(2.6210\) \(\Gamma_0(N)\)-optimal