# Properties

 Label 20286.k Number of curves $6$ Conductor $20286$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("20286.k1")

sage: E.isogeny_class()

## Elliptic curves in class 20286.k

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
20286.k1 20286be5 [1, -1, 0, -34898103, -79341193359] [2] 1572864
20286.k2 20286be4 [1, -1, 0, -7811883, 8405643861] [2] 786432
20286.k3 20286be3 [1, -1, 0, -2237643, -1171648395] [2, 2] 786432
20286.k4 20286be2 [1, -1, 0, -508923, 119705445] [2, 2] 393216
20286.k5 20286be1 [1, -1, 0, 55557, 10309221] [2] 196608 $$\Gamma_0(N)$$-optimal
20286.k6 20286be6 [1, -1, 0, 2763297, -5669493831] [2] 1572864

## Rank

sage: E.rank()

The elliptic curves in class 20286.k have rank $$0$$.

## Modular form 20286.2.a.k

sage: E.q_eigenform(10)

$$q - q^{2} + q^{4} - 2q^{5} - q^{8} + 2q^{10} + 4q^{11} + 2q^{13} + q^{16} - 6q^{17} - 4q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.