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Properties

Label 66066.bg
Number of curves $4$
Conductor $66066$
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, 0, 1, -50460, -4366892]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 0, 1, -50460, -4366892]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 0, 1, -50460, -4366892]) 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 66066.bg have rank \(0\).

L-function data

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

Complex multiplication

The elliptic curves in class 66066.bg do not have complex multiplication.

Modular form 66066.2.a.bg

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);
 
\(q - q^{2} + q^{3} + q^{4} + 2 q^{5} - q^{6} - q^{7} - q^{8} + q^{9} - 2 q^{10} + q^{12} + q^{13} + q^{14} + 2 q^{15} + q^{16} - 6 q^{17} - q^{18} + 4 q^{19} + O(q^{20})\) 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 66066.bg

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
66066.bg1 66066w4 \([1, 0, 1, -50460, -4366892]\) \(8020417344913/187278\) \(331774400958\) \([2]\) \(245760\) \(1.3229\)  
66066.bg2 66066w2 \([1, 0, 1, -3270, -63164]\) \(2181825073/298116\) \(528130679076\) \([2, 2]\) \(122880\) \(0.97635\)  
66066.bg3 66066w1 \([1, 0, 1, -850, 8468]\) \(38272753/4368\) \(7738178448\) \([2]\) \(61440\) \(0.62977\) \(\Gamma_0(N)\)-optimal
66066.bg4 66066w3 \([1, 0, 1, 5200, -334204]\) \(8780064047/32388174\) \(-57377625919614\) \([2]\) \(245760\) \(1.3229\)