# Properties

 Label 221991e Number of curves 6 Conductor 221991 CM no Rank 1 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("221991.h1")

sage: E.isogeny_class()

## Elliptic curves in class 221991e

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
221991.h4 221991e1 [1, 0, 0, -32694, -2277357] [2] 576000 $$\Gamma_0(N)$$-optimal
221991.h3 221991e2 [1, 0, 0, -37499, -1565256] [2, 2] 1152000
221991.h2 221991e3 [1, 0, 0, -272944, 53764319] [2, 2] 2304000
221991.h6 221991e4 [1, 0, 0, 121066, -11301147] [2] 2304000
221991.h1 221991e5 [1, 0, 0, -4342779, 3483007290] [2] 4608000
221991.h5 221991e6 [1, 0, 0, 29771, 166555928] [2] 4608000

## Rank

sage: E.rank()

The elliptic curves in class 221991e have rank $$1$$.

## Modular form 221991.2.a.h

sage: E.q_eigenform(10)

$$q - q^{2} + q^{3} - q^{4} - 2q^{5} - q^{6} + q^{7} + 3q^{8} + q^{9} + 2q^{10} + q^{11} - q^{12} - 6q^{13} - q^{14} - 2q^{15} - q^{16} - 2q^{17} - q^{18} + 4q^{19} + O(q^{20})$$

## 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.