# Properties

 Label 139650jk Number of curves $4$ Conductor $139650$ CM no Rank $1$ Graph

# Related objects

Show commands: SageMath
sage: E = EllipticCurve("jk1")

sage: E.isogeny_class()

## Elliptic curves in class 139650jk

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
139650.cc3 139650jk1 $$[1, 1, 0, -38000, -2850000]$$ $$3301293169/22800$$ $$41912456250000$$ $$[2]$$ $$589824$$ $$1.4474$$ $$\Gamma_0(N)$$-optimal
139650.cc2 139650jk2 $$[1, 1, 0, -62500, 1241500]$$ $$14688124849/8122500$$ $$14931312539062500$$ $$[2, 2]$$ $$1179648$$ $$1.7940$$
139650.cc1 139650jk3 $$[1, 1, 0, -760750, 254706250]$$ $$26487576322129/44531250$$ $$81860266113281250$$ $$[2]$$ $$2359296$$ $$2.1406$$
139650.cc4 139650jk4 $$[1, 1, 0, 243750, 10122750]$$ $$871257511151/527800050$$ $$-970236688788281250$$ $$[2]$$ $$2359296$$ $$2.1406$$

## Rank

sage: E.rank()

The elliptic curves in class 139650jk have rank $$1$$.

## Complex multiplication

The elliptic curves in class 139650jk do not have complex multiplication.

## Modular form 139650.2.a.jk

sage: E.q_eigenform(10)

$$q - q^{2} - q^{3} + q^{4} + q^{6} - q^{8} + q^{9} + 4q^{11} - q^{12} + 2q^{13} + q^{16} + 2q^{17} - q^{18} + q^{19} + O(q^{20})$$

## 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 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.