# Properties

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

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("439569.z1")

sage: E.isogeny_class()

## Elliptic curves in class 439569z

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
439569.z4 439569z1 [1, -1, 1, -236936849, -1403712409264] [2] 49545216 $$\Gamma_0(N)$$-optimal*
439569.z3 439569z2 [1, -1, 1, -239134694, -1376341326772] [2, 2] 99090432 $$\Gamma_0(N)$$-optimal*
439569.z2 439569z3 [1, -1, 1, -610570499, 3938905042778] [2, 2] 198180864 $$\Gamma_0(N)$$-optimal*
439569.z5 439569z4 [1, -1, 1, 97135591, -4939864790974] [2] 198180864
439569.z1 439569z5 [1, -1, 1, -8867874164, 321376081297106] [2] 396361728 $$\Gamma_0(N)$$-optimal*
439569.z6 439569z6 [1, -1, 1, 1703760286, 26673039209990] [2] 396361728
*optimality has not been proved rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 4 curves highlighted, and conditionally curve 439569z1.

## Rank

sage: E.rank()

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

## 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 Cremona numbering.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.