# Properties

 Label 139650.dn Number of curves $2$ Conductor $139650$ CM no Rank $0$ Graph

# Learn more

Show commands: SageMath
sage: E = EllipticCurve("dn1")

sage: E.isogeny_class()

## Elliptic curves in class 139650.dn

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
139650.dn1 139650gr1 $$[1, 0, 1, -32903526, -72140372552]$$ $$6248109436056487/50429952000$$ $$31797351000576000000000$$ $$[2]$$ $$18579456$$ $$3.1450$$ $$\Gamma_0(N)$$-optimal
139650.dn2 139650gr2 $$[1, 0, 1, -10951526, -166885204552]$$ $$-230380217865127/18948168000000$$ $$-11947295700655875000000000$$ $$[2]$$ $$37158912$$ $$3.4915$$

## Rank

sage: E.rank()

The elliptic curves in class 139650.dn have rank $$0$$.

## Complex multiplication

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

## Modular form 139650.2.a.dn

sage: E.q_eigenform(10)

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

## 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 LMFDB numbering.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.