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Properties

Label 6150.k
Number of curves $4$
Conductor $6150$
CM no
Rank $1$
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, -318900, -69408000]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, 1, 0, -318900, -69408000]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, 1, 0, -318900, -69408000]) 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 6150.k have rank \(1\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 + T\)
\(3\)\(1 + T\)
\(5\)\(1\)
\(41\)\(1 + T\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 - 4 T + 7 T^{2}\) 1.7.ae
\(11\) \( 1 + 6 T + 11 T^{2}\) 1.11.g
\(13\) \( 1 - 4 T + 13 T^{2}\) 1.13.ae
\(17\) \( 1 + 17 T^{2}\) 1.17.a
\(19\) \( 1 - 2 T + 19 T^{2}\) 1.19.ac
\(23\) \( 1 + 23 T^{2}\) 1.23.a
\(29\) \( 1 + 6 T + 29 T^{2}\) 1.29.g
$\cdots$$\cdots$$\cdots$
 
See L-function page for more information

Complex multiplication

The elliptic curves in class 6150.k do not have complex multiplication.

Modular form 6150.2.a.k

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);
 
\(q - q^{2} - q^{3} + q^{4} + q^{6} + 4 q^{7} - q^{8} + q^{9} - 6 q^{11} - q^{12} + 4 q^{13} - 4 q^{14} + q^{16} - q^{18} + 2 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)\)

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 6150.k

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
6150.k1 6150c3 \([1, 1, 0, -318900, -69408000]\) \(229545811016693569/155072250000\) \(2423003906250000\) \([2]\) \(82944\) \(1.8901\)  
6150.k2 6150c4 \([1, 1, 0, -256400, -97345500]\) \(-119305480789133569/192379221760500\) \(-3005925340007812500\) \([2]\) \(165888\) \(2.2367\)  
6150.k3 6150c1 \([1, 1, 0, -12900, 450000]\) \(15195864748609/3060633600\) \(47822400000000\) \([2]\) \(27648\) \(1.3408\) \(\Gamma_0(N)\)-optimal
6150.k4 6150c2 \([1, 1, 0, 27100, 2730000]\) \(140859621945791/285872742720\) \(-4466761605000000\) \([2]\) \(55296\) \(1.6874\)