# Properties

 Label 226941b Number of curves $6$ Conductor $226941$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("226941.i1")

sage: E.isogeny_class()

## Elliptic curves in class 226941b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
226941.i5 226941b1 [1, 0, 0, -12707, -791928] [2] 720896 $$\Gamma_0(N)$$-optimal
226941.i4 226941b2 [1, 0, 0, -226952, -41627025] [2, 2] 1441792
226941.i3 226941b3 [1, 0, 0, -250757, -32366880] [2, 2] 2883584
226941.i1 226941b4 [1, 0, 0, -3631067, -2663476398] [2] 2883584
226941.i2 226941b5 [1, 0, 0, -1591772, 748372053] [2] 5767168
226941.i6 226941b6 [1, 0, 0, 709378, -220361313] [2] 5767168

## Rank

sage: E.rank()

The elliptic curves in class 226941b have rank $$1$$.

## Modular form 226941.2.a.i

sage: E.q_eigenform(10)

$$q - q^{2} + q^{3} - q^{4} + 2q^{5} - q^{6} + 3q^{8} + q^{9} - 2q^{10} + q^{11} - q^{12} + q^{13} + 2q^{15} - q^{16} + 6q^{17} - q^{18} + 4q^{19} + O(q^{20})$$

## 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.