# Properties

 Label 22050.ej Number of curves $8$ Conductor $22050$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("22050.ej1")

sage: E.isogeny_class()

## Elliptic curves in class 22050.ej

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
22050.ej1 22050dz8 [1, -1, 1, -71122505, 145263350747] [2] 5308416
22050.ej2 22050dz5 [1, -1, 1, -63515255, 194850052247] [2] 1769472
22050.ej3 22050dz6 [1, -1, 1, -29778755, -60876586753] [2, 2] 2654208
22050.ej4 22050dz3 [1, -1, 1, -29558255, -61846345753] [2] 1327104
22050.ej5 22050dz2 [1, -1, 1, -3980255, 3028282247] [2, 2] 884736
22050.ej6 22050dz4 [1, -1, 1, -893255, 7603216247] [2] 1769472
22050.ej7 22050dz1 [1, -1, 1, -452255, -41077753] [2] 442368 $$\Gamma_0(N)$$-optimal
22050.ej8 22050dz7 [1, -1, 1, 8036995, -204954594253] [2] 5308416

## Rank

sage: E.rank()

The elliptic curves in class 22050.ej have rank $$0$$.

## Modular form 22050.2.a.ej

sage: E.q_eigenform(10)

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

## 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.

$$\left(\begin{array}{rrrrrrrr} 1 & 3 & 2 & 4 & 6 & 12 & 12 & 4 \\ 3 & 1 & 6 & 12 & 2 & 4 & 4 & 12 \\ 2 & 6 & 1 & 2 & 3 & 6 & 6 & 2 \\ 4 & 12 & 2 & 1 & 6 & 12 & 3 & 4 \\ 6 & 2 & 3 & 6 & 1 & 2 & 2 & 6 \\ 12 & 4 & 6 & 12 & 2 & 1 & 4 & 3 \\ 12 & 4 & 6 & 3 & 2 & 4 & 1 & 12 \\ 4 & 12 & 2 & 4 & 6 & 3 & 12 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.