# Properties

 Label 439569.z Number of curves $6$ Conductor $439569$ CM no Rank $2$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("z1")

sage: E.isogeny_class()

## Elliptic curves in class 439569.z

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
439569.z1 439569z5 $$[1, -1, 1, -8867874164, 321376081297106]$$ $$908031902324522977/161726530797$$ $$13736068281063240698341006173$$ $$[2]$$ $$396361728$$ $$4.4035$$ $$\Gamma_0(N)$$-optimal*
439569.z2 439569z3 $$[1, -1, 1, -610570499, 3938905042778]$$ $$296380748763217/92608836489$$ $$7865631539697392974191001401$$ $$[2, 2]$$ $$198180864$$ $$4.0570$$ $$\Gamma_0(N)$$-optimal*
439569.z3 439569z2 $$[1, -1, 1, -239134694, -1376341326772]$$ $$17806161424897/668584449$$ $$56785498321536990586453041$$ $$[2, 2]$$ $$99090432$$ $$3.7104$$ $$\Gamma_0(N)$$-optimal*
439569.z4 439569z1 $$[1, -1, 1, -236936849, -1403712409264]$$ $$17319700013617/25857$$ $$2196136377829484881713$$ $$[2]$$ $$49545216$$ $$3.3638$$ $$\Gamma_0(N)$$-optimal*
439569.z5 439569z4 $$[1, -1, 1, 97135591, -4939864790974]$$ $$1193377118543/124806800313$$ $$-10600330833734758092416243817$$ $$[2]$$ $$198180864$$ $$4.0570$$
439569.z6 439569z6 $$[1, -1, 1, 1703760286, 26673039209990]$$ $$6439735268725823/7345472585373$$ $$-623879783311553616901620729357$$ $$[2]$$ $$396361728$$ $$4.4035$$
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 4 curves highlighted, and conditionally curve 439569.z1.

## Rank

sage: E.rank()

The elliptic curves in class 439569.z have rank $$2$$.

## Complex multiplication

The elliptic curves in class 439569.z do not have complex multiplication.

## Modular form 439569.2.a.z

sage: E.q_eigenform(10)

$$q - q^{2} - q^{4} + 2q^{5} + 3q^{8} - 2q^{10} - 4q^{11} - q^{16} + 4q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 8 & 8 & 4 \\ 2 & 1 & 2 & 4 & 4 & 2 \\ 4 & 2 & 1 & 2 & 2 & 4 \\ 8 & 4 & 2 & 1 & 4 & 8 \\ 8 & 4 & 2 & 4 & 1 & 8 \\ 4 & 2 & 4 & 8 & 8 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.