# Properties

 Label 201810.w Number of curves $4$ Conductor $201810$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("w1")

sage: E.isogeny_class()

## Elliptic curves in class 201810.w

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
201810.w1 201810cb4 [1, 0, 1, -358954, -82806034] [2] 1843200
201810.w2 201810cb2 [1, 0, 1, -22604, -1274794] [2, 2] 921600
201810.w3 201810cb1 [1, 0, 1, -3384, 47542] [2] 460800 $$\Gamma_0(N)$$-optimal
201810.w4 201810cb3 [1, 0, 1, 6226, -4296178] [2] 1843200

## Rank

sage: E.rank()

The elliptic curves in class 201810.w have rank $$1$$.

## Complex multiplication

The elliptic curves in class 201810.w do not have complex multiplication.

## Modular form 201810.2.a.w

sage: E.q_eigenform(10)

$$q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} - q^{7} - q^{8} + q^{9} + q^{10} + 4q^{11} + q^{12} + 2q^{13} + q^{14} - q^{15} + q^{16} + 6q^{17} - q^{18} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.