Properties

Label 84150.df
Number of curves $8$
Conductor $84150$
CM no
Rank $0$
Graph

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Show commands: SageMath
sage: E = EllipticCurve("df1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 84150.df

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
84150.df1 84150bk8 \([1, -1, 0, -45278208567, 3708371995205341]\) \(901247067798311192691198986281/552431869440\) \(6292544262840000000\) \([2]\) \(127401984\) \(4.3148\)  
84150.df2 84150bk7 \([1, -1, 0, -2848896567, 57125962949341]\) \(224494757451893010998773801/6152490825146276160000\) \(70080715805181801885000000000\) \([2]\) \(127401984\) \(4.3148\)  
84150.df3 84150bk6 \([1, -1, 0, -2829888567, 57943820165341]\) \(220031146443748723000125481/172266701724057600\) \(1962225399325593600000000\) \([2, 2]\) \(63700992\) \(3.9682\)  
84150.df4 84150bk5 \([1, -1, 0, -559104192, 5084876362216]\) \(1696892787277117093383481/1440538624914939000\) \(16408635274421727046875000\) \([2]\) \(42467328\) \(3.7655\)  
84150.df5 84150bk4 \([1, -1, 0, -366162192, -2667956503784]\) \(476646772170172569823801/5862293314453125000\) \(66775184784942626953125000\) \([2]\) \(42467328\) \(3.7655\)  
84150.df6 84150bk3 \([1, -1, 0, -175680567, 918161285341]\) \(-52643812360427830814761/1504091705903677440\) \(-17132544587559075840000000\) \([2]\) \(31850496\) \(3.6217\)  
84150.df7 84150bk2 \([1, -1, 0, -42729192, 41441737216]\) \(757443433548897303481/373234243041000000\) \(4251371299638890625000000\) \([2, 2]\) \(21233664\) \(3.4189\)  
84150.df8 84150bk1 \([1, -1, 0, 9758808, 4962577216]\) \(9023321954633914439/6156756739584000\) \(-70129307236824000000000\) \([2]\) \(10616832\) \(3.0724\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 84150.df have rank \(0\).

Complex multiplication

The elliptic curves in class 84150.df do not have complex multiplication.

Modular form 84150.2.a.df

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

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

Isogeny graph

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

The vertices are labelled with LMFDB labels.