Properties

Label 47040fk
Number of curves $4$
Conductor $47040$
CM no
Rank $0$
Graph

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Show commands for: SageMath
sage: E = EllipticCurve("fk1")
 
sage: E.isogeny_class()
 

Elliptic curves in class 47040fk

sage: E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
47040.dk4 47040fk1 \([0, -1, 0, -78407660, -263920143150]\) \(7079962908642659949376/100085966990454375\) \(753600891549437872920000\) \([2]\) \(10321920\) \(3.3864\) \(\Gamma_0(N)\)-optimal
47040.dk2 47040fk2 \([0, -1, 0, -1250235065, -17014724166663]\) \(448487713888272974160064/91549016015625\) \(44116583158670400000000\) \([2, 2]\) \(20643840\) \(3.7329\)  
47040.dk3 47040fk3 \([0, -1, 0, -1245948545, -17137192614975]\) \(-55486311952875723077768/801237030029296875\) \(-3088866847815000000000000000\) \([2]\) \(41287680\) \(4.0795\)  
47040.dk1 47040fk4 \([0, -1, 0, -20003760065, -1088962462461663]\) \(229625675762164624948320008/9568125\) \(36886293319680000\) \([2]\) \(41287680\) \(4.0795\)  

Rank

sage: E.rank()
 

The elliptic curves in class 47040fk have rank \(0\).

Complex multiplication

The elliptic curves in class 47040fk do not have complex multiplication.

Modular form 47040.2.a.fk

sage: E.q_eigenform(10)
 
\(q - q^{3} + q^{5} + q^{9} + 4q^{11} - 6q^{13} - q^{15} - 6q^{17} - 4q^{19} + O(q^{20})\)  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.

\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)

Isogeny graph

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

The vertices are labelled with Cremona labels.