↑

Properties

Label 150c
Number of curves $8$
Conductor $150$
CM no
Rank $0$
Graph

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the isogeny class
 
Copy content sage:E = EllipticCurve([1, 1, 1, 37, 281]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 1, 1, 37, 281]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 1, 1, 37, 281]) 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 150c have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(3\)\(1 - T\)
\(5\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 + 2 T + 7 T^{2}\) 1.7.c
\(11\) \( 1 - 2 T + 11 T^{2}\) 1.11.ac
\(13\) \( 1 + 6 T + 13 T^{2}\) 1.13.g
\(17\) \( 1 + 2 T + 17 T^{2}\) 1.17.c
\(19\) \( 1 + 19 T^{2}\) 1.19.a
\(23\) \( 1 - 4 T + 23 T^{2}\) 1.23.ae
\(29\) \( 1 + 29 T^{2}\) 1.29.a
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 150c do not have complex multiplication.

Modular form 150.2.a.c

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);
 
\(q + q^{2} - q^{3} + q^{4} - q^{6} + 4 q^{7} + q^{8} + q^{9} - q^{12} - 2 q^{13} + 4 q^{14} + q^{16} - 6 q^{17} + q^{18} - 4 q^{19} + O(q^{20})\) 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 Cremona numbering.

\(\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

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

Elliptic curves in class 150c

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
150.b8 150c1 \([1, 1, 1, 37, 281]\) \(357911/2160\) \(-33750000\) \([4]\) \(48\) \(0.12643\) \(\Gamma_0(N)\)-optimal
150.b6 150c2 \([1, 1, 1, -463, 3281]\) \(702595369/72900\) \(1139062500\) \([2, 2]\) \(96\) \(0.47301\)  
150.b7 150c3 \([1, 1, 1, -338, -7969]\) \(-273359449/1536000\) \(-24000000000\) \([4]\) \(144\) \(0.67574\)  
150.b5 150c4 \([1, 1, 1, -1713, -24219]\) \(35578826569/5314410\) \(83037656250\) \([2]\) \(192\) \(0.81958\)  
150.b4 150c5 \([1, 1, 1, -7213, 232781]\) \(2656166199049/33750\) \(527343750\) \([2]\) \(192\) \(0.81958\)  
150.b3 150c6 \([1, 1, 1, -8338, -295969]\) \(4102915888729/9000000\) \(140625000000\) \([2, 2]\) \(288\) \(1.0223\)  
150.b1 150c7 \([1, 1, 1, -133338, -18795969]\) \(16778985534208729/81000\) \(1265625000\) \([2]\) \(576\) \(1.3689\)  
150.b2 150c8 \([1, 1, 1, -11338, -67969]\) \(10316097499609/5859375000\) \(91552734375000\) \([2]\) \(576\) \(1.3689\)