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Properties

Label 450.f
Number of curves $4$
Conductor $450$
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, -186305, -30804303]) E.isogeny_class()
 
Copy content magma:E := EllipticCurve([1, -1, 1, -186305, -30804303]); IsogenousCurves(E);
 
Copy content gp:E = ellinit([1, -1, 1, -186305, -30804303]) 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 450.f have rank \(0\).

L-function data

Bad L-factors:
Prime L-Factor
\(2\)\(1 - T\)
\(3\)\(1\)
\(5\)\(1\)
 
Good L-factors:
Prime L-Factor Isogeny Class over \(\mathbb{F}_p\)
\(7\) \( 1 - 2 T + 7 T^{2}\) 1.7.ac
\(11\) \( 1 + 2 T + 11 T^{2}\) 1.11.c
\(13\) \( 1 - 6 T + 13 T^{2}\) 1.13.ag
\(17\) \( 1 + 2 T + 17 T^{2}\) 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 450.f do not have complex multiplication.

Modular form 450.2.a.f

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

\(\left(\begin{array}{rrrr} 1 & 5 & 2 & 10 \\ 5 & 1 & 10 & 2 \\ 2 & 10 & 1 & 5 \\ 10 & 2 & 5 & 1 \end{array}\right)\)

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 450.f

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
450.f1 450a4 \([1, -1, 1, -186305, -30804303]\) \(502270291349/1889568\) \(2690420062500000\) \([2]\) \(3200\) \(1.8200\)  
450.f2 450a2 \([1, -1, 1, -11930, 504447]\) \(131872229/18\) \(25628906250\) \([2]\) \(640\) \(1.0153\)  
450.f3 450a3 \([1, -1, 1, -6305, -924303]\) \(-19465109/248832\) \(-354294000000000\) \([2]\) \(1600\) \(1.4735\)  
450.f4 450a1 \([1, -1, 1, -680, 9447]\) \(-24389/12\) \(-17085937500\) \([2]\) \(320\) \(0.66875\) \(\Gamma_0(N)\)-optimal