# Properties

 Label 22050bd Number of curves 8 Conductor 22050 CM no Rank 1 Graph

# Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("22050.bo1")

sage: E.isogeny_class()

## Elliptic curves in class 22050bd

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
22050.bo7 22050bd1 [1, -1, 0, -5485167, 4944093741] [2] 884736 $$\Gamma_0(N)$$-optimal
22050.bo6 22050bd2 [1, -1, 0, -6367167, 3248007741] [2, 2] 1769472
22050.bo5 22050bd3 [1, -1, 0, -16234542, -19130371884] [2] 2654208
22050.bo8 22050bd4 [1, -1, 0, 21195333, 23837195241] [2] 3538944
22050.bo4 22050bd5 [1, -1, 0, -48041667, -125901267759] [2] 3538944
22050.bo2 22050bd6 [1, -1, 0, -242026542, -1449071107884] [2, 2] 5308416
22050.bo3 22050bd7 [1, -1, 0, -224386542, -1669271227884] [2] 10616832
22050.bo1 22050bd8 [1, -1, 0, -3872338542, -92747787595884] [2] 10616832

## Rank

sage: E.rank()

The elliptic curves in class 22050bd have rank $$1$$.

## Modular form 22050.2.a.bo

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.