# Properties

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

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("dk1")

sage: E.isogeny_class()

## Elliptic curves in class 47040.dk

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
47040.dk1 47040fk4 $$[0, -1, 0, -20003760065, -1088962462461663]$$ $$229625675762164624948320008/9568125$$ $$36886293319680000$$ $$[2]$$ $$41287680$$ $$4.0795$$
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.dk4 47040fk1 $$[0, -1, 0, -78407660, -263920143150]$$ $$7079962908642659949376/100085966990454375$$ $$753600891549437872920000$$ $$[2]$$ $$10321920$$ $$3.3864$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 47040.dk have rank $$0$$.

## Complex multiplication

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

## Modular form 47040.2.a.dk

sage: E.q_eigenform(10)

$$q - q^{3} + q^{5} + q^{9} + 4q^{11} - 6q^{13} - q^{15} - 6q^{17} - 4q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.