# Properties

 Label 123840.dk Number of curves $4$ Conductor $123840$ CM no Rank $1$ Graph

# Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 123840.dk

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
123840.dk1 123840cd4 $$[0, 0, 0, -482988, -128316112]$$ $$65202655558249/512820150$$ $$98001456817766400$$ $$[2]$$ $$1179648$$ $$2.0885$$
123840.dk2 123840cd2 $$[0, 0, 0, -50988, 1111088]$$ $$76711450249/41602500$$ $$7950361559040000$$ $$[2, 2]$$ $$589824$$ $$1.7419$$
123840.dk3 123840cd1 $$[0, 0, 0, -39468, 3014192]$$ $$35578826569/51600$$ $$9860913561600$$ $$[2]$$ $$294912$$ $$1.3954$$ $$\Gamma_0(N)$$-optimal
123840.dk4 123840cd3 $$[0, 0, 0, 196692, 8739632]$$ $$4403686064471/2721093750$$ $$-520009113600000000$$ $$[2]$$ $$1179648$$ $$2.0885$$

## Rank

sage: E.rank()

The elliptic curves in class 123840.dk have rank $$1$$.

## Complex multiplication

The elliptic curves in class 123840.dk do not have complex multiplication.

## Modular form 123840.2.a.dk

sage: E.q_eigenform(10)

$$q - q^{5} + 4q^{7} + 2q^{13} - 2q^{17} - 4q^{19} + O(q^{20})$$

## 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.