# Properties

 Label 36822i Number of curves 6 Conductor 36822 CM no Rank 1 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("36822.e1")

sage: E.isogeny_class()

## Elliptic curves in class 36822i

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
36822.e5 36822i1 [1, 1, 0, -12281, -490971] [2] 110592 $$\Gamma_0(N)$$-optimal
36822.e4 36822i2 [1, 1, 0, -41161, 2633845] [2, 2] 221184
36822.e6 36822i3 [1, 1, 0, 81579, 15472449] [2] 442368
36822.e2 36822i4 [1, 1, 0, -625981, 190361065] [2, 2] 442368
36822.e3 36822i5 [1, 1, 0, -593491, 211044199] [2] 884736
36822.e1 36822i6 [1, 1, 0, -10015591, 12195916411] [2] 884736

## Rank

sage: E.rank()

The elliptic curves in class 36822i have rank $$1$$.

## Modular form 36822.2.a.e

sage: E.q_eigenform(10)

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

## 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 Cremona numbering.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.