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Properties

Label 84966.f
Number of curves $2$
Conductor $84966$
CM no
Rank $1$
Graph

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

Elliptic curves in class 84966.f

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
84966.f1 84966f2 \([1, 1, 0, -6538480939, -58252741166339]\) \(222165413800219579417/118033833938006016\) \(16424205199578486928066285154304\) \([]\) \(365783040\) \(4.6811\)  
84966.f2 84966f1 \([1, 1, 0, -3768376924, 89035255022176]\) \(42531320912955257257/1127938881456\) \(156950757452701066497887664\) \([]\) \(121927680\) \(4.1318\) \(\Gamma_0(N)\)-optimal

Rank

sage: E.rank()
 

The elliptic curves in class 84966.f have rank \(1\).

Complex multiplication

The elliptic curves in class 84966.f do not have complex multiplication.

Modular form 84966.2.a.f

sage: E.q_eigenform(10)
 
\(q - q^{2} - q^{3} + q^{4} - 3 q^{5} + q^{6} - q^{8} + q^{9} + 3 q^{10} + 6 q^{11} - q^{12} + 5 q^{13} + 3 q^{15} + q^{16} - q^{18} - 4 q^{19} + O(q^{20})\) 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.

\(\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with LMFDB labels.