Properties

Label 54450.ef
Number of curves $8$
Conductor $54450$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("ef1")
 
E.isogeny_class()
 

Elliptic curves in class 54450.ef

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
54450.ef1 54450gf7 \([1, -1, 1, -145205105, -673437860103]\) \(16778985534208729/81000\) \(1634514148265625000\) \([2]\) \(6635520\) \(3.1171\)  
54450.ef2 54450gf8 \([1, -1, 1, -12347105, -2269736103]\) \(10316097499609/5859375000\) \(118237423919677734375000\) \([2]\) \(6635520\) \(3.1171\)  
54450.ef3 54450gf6 \([1, -1, 1, -9080105, -10509110103]\) \(4102915888729/9000000\) \(181612683140625000000\) \([2, 2]\) \(3317760\) \(2.7706\)  
54450.ef4 54450gf5 \([1, -1, 1, -7854980, 8475426897]\) \(2656166199049/33750\) \(681047561777343750\) \([2]\) \(2211840\) \(2.5678\)  
54450.ef5 54450gf4 \([1, -1, 1, -1865480, -844235103]\) \(35578826569/5314410\) \(107240473267707656250\) \([2]\) \(2211840\) \(2.5678\)  
54450.ef6 54450gf2 \([1, -1, 1, -504230, 124974897]\) \(702595369/72900\) \(1471062733439062500\) \([2, 2]\) \(1105920\) \(2.2213\)  
54450.ef7 54450gf3 \([1, -1, 1, -368105, -281222103]\) \(-273359449/1536000\) \(-30995231256000000000\) \([2]\) \(1658880\) \(2.4240\)  
54450.ef8 54450gf1 \([1, -1, 1, 40270, 9540897]\) \(357911/2160\) \(-43587043953750000\) \([2]\) \(552960\) \(1.8747\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 54450.ef have rank \(0\).

Complex multiplication

The elliptic curves in class 54450.ef do not have complex multiplication.

Modular form 54450.2.a.ef

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.