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Properties

Label 18050y
Number of curves $4$
Conductor $18050$
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, 1, -188, 11631]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 1, 1, -188, 11631]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 1, 1, -188, 11631]) 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 18050y have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(5\)\(1\)
\(19\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(3\) \( 1 - 3 T + 3 T^{2}\) 1.3.ad
\(7\) \( 1 - 3 T + 7 T^{2}\) 1.7.ad
\(11\) \( 1 + 2 T + 11 T^{2}\) 1.11.c
\(13\) \( 1 + 3 T + 13 T^{2}\) 1.13.d
\(17\) \( 1 - T + 17 T^{2}\) 1.17.ab
\(23\) \( 1 + 5 T + 23 T^{2}\) 1.23.f
\(29\) \( 1 - 3 T + 29 T^{2}\) 1.29.ad
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 18050y do not have complex multiplication.

Modular form 18050.2.a.y

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);
 
\(q + q^{2} - q^{3} + q^{4} - q^{6} + 2 q^{7} + q^{8} - 2 q^{9} - 3 q^{11} - q^{12} + 4 q^{13} + 2 q^{14} + q^{16} - 3 q^{17} - 2 q^{18} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrr} 1 & 3 & 5 & 15 \\ 3 & 1 & 15 & 5 \\ 5 & 15 & 1 & 3 \\ 15 & 5 & 3 & 1 \end{array}\right)\)

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 18050y

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
18050.o3 18050y1 \([1, 1, 1, -188, 11631]\) \(-25/2\) \(-58807351250\) \([]\) \(13500\) \(0.74612\) \(\Gamma_0(N)\)-optimal
18050.o1 18050y2 \([1, 1, 1, -45313, 3693831]\) \(-349938025/8\) \(-235229405000\) \([]\) \(40500\) \(1.2954\)  
18050.o2 18050y3 \([1, 1, 1, -27263, -2100219]\) \(-121945/32\) \(-588073512500000\) \([]\) \(67500\) \(1.5508\)  
18050.o4 18050y4 \([1, 1, 1, 198362, 15498531]\) \(46969655/32768\) \(-602187276800000000\) \([]\) \(202500\) \(2.1001\)