Properties

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

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Show commands: SageMath
E = EllipticCurve("e1")
 
E.isogeny_class()
 

Elliptic curves in class 450.e

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
450.e1 450e4 \([1, -1, 1, -28730, -1867103]\) \(8527173507/200\) \(61509375000\) \([2]\) \(1152\) \(1.1822\)  
450.e2 450e3 \([1, -1, 1, -1730, -31103]\) \(-1860867/320\) \(-98415000000\) \([2]\) \(576\) \(0.83566\)  
450.e3 450e2 \([1, -1, 1, -605, 1647]\) \(57960603/31250\) \(13183593750\) \([2]\) \(384\) \(0.63293\)  
450.e4 450e1 \([1, -1, 1, 145, 147]\) \(804357/500\) \(-210937500\) \([2]\) \(192\) \(0.28636\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

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

Complex multiplication

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

Modular form 450.2.a.e

sage: E.q_eigenform(10)
 
\(q + q^{2} + q^{4} - 2 q^{7} + q^{8} + 6 q^{11} + 4 q^{13} - 2 q^{14} + q^{16} + 6 q^{17} - 4 q^{19} + O(q^{20})\) 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.

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.