Properties

Label 450.f
Number of curves $4$
Conductor $450$
CM no
Rank $0$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("f1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 450.f

sage: E.isogeny_class().curves
 
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

Rank

sage: E.rank()
 

The elliptic curves in class 450.f have rank \(0\).

Complex multiplication

The elliptic curves in class 450.f do not have complex multiplication.

Modular form 450.2.a.f

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{4} + 2 q^{7} + q^{8} - 2 q^{11} + 6 q^{13} + 2 q^{14} + q^{16} - 2 q^{17} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

sage: E.isogeny_class().matrix()
 

The \(i,j\) 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

sage: E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with LMFDB labels.