# Properties

 Label 418950.kk Number of curves $4$ Conductor $418950$ CM no Rank $1$ Graph

# Learn more

Show commands: SageMath
sage: E = EllipticCurve("kk1")

sage: E.isogeny_class()

## Elliptic curves in class 418950.kk

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
418950.kk1 418950kk4 $$[1, -1, 1, -6846755, -6883915503]$$ $$26487576322129/44531250$$ $$59676133996582031250$$ $$[2]$$ $$18874368$$ $$2.6899$$
418950.kk2 418950kk2 $$[1, -1, 1, -562505, -34083003]$$ $$14688124849/8122500$$ $$10884926840976562500$$ $$[2, 2]$$ $$9437184$$ $$2.3433$$
418950.kk3 418950kk1 $$[1, -1, 1, -342005, 76607997]$$ $$3301293169/22800$$ $$30554180606250000$$ $$[2]$$ $$4718592$$ $$1.9967$$ $$\Gamma_0(N)$$-optimal*
418950.kk4 418950kk3 $$[1, -1, 1, 2193745, -271120503]$$ $$871257511151/527800050$$ $$-707302546126657031250$$ $$[2]$$ $$18874368$$ $$2.6899$$
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 0 curves highlighted, and conditionally curve 418950.kk1.

## Rank

sage: E.rank()

The elliptic curves in class 418950.kk have rank $$1$$.

## Complex multiplication

The elliptic curves in class 418950.kk do not have complex multiplication.

## Modular form 418950.2.a.kk

sage: E.q_eigenform(10)

$$q + q^{2} + q^{4} + q^{8} - 4q^{11} + 2q^{13} + q^{16} - 2q^{17} + q^{19} + O(q^{20})$$

## 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.