Properties

Label 76050dk
Number of curves $4$
Conductor $76050$
CM no
Rank $0$
Graph

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Show commands: SageMath
E = EllipticCurve("dk1")
 
E.isogeny_class()
 

Elliptic curves in class 76050dk

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
76050.fo3 76050dk1 \([1, -1, 1, -32480, 2574147]\) \(-1860867/320\) \(-651619215000000\) \([2]\) \(414720\) \(1.5688\) \(\Gamma_0(N)\)-optimal
76050.fo2 76050dk2 \([1, -1, 1, -539480, 152646147]\) \(8527173507/200\) \(407262009375000\) \([2]\) \(829440\) \(1.9154\)  
76050.fo4 76050dk3 \([1, -1, 1, 221020, -10945853]\) \(804357/500\) \(-742235012085937500\) \([2]\) \(1244160\) \(2.1181\)  
76050.fo1 76050dk4 \([1, -1, 1, -919730, -88516853]\) \(57960603/31250\) \(46389688255371093750\) \([2]\) \(2488320\) \(2.4647\)  

Rank

sage: E.rank()
 

The elliptic curves in class 76050dk have rank \(0\).

Complex multiplication

The elliptic curves in class 76050dk do not have complex multiplication.

Modular form 76050.2.a.dk

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