# Properties

 Label 22050es Number of curves $4$ Conductor $22050$ CM no Rank $0$ Graph

# Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("22050.dd1")

sage: E.isogeny_class()

## Elliptic curves in class 22050es

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
22050.dd4 22050es1 [1, -1, 1, 25495, -2502503] [2] 147456 $$\Gamma_0(N)$$-optimal
22050.dd3 22050es2 [1, -1, 1, -195005, -26757503] [2, 2] 294912
22050.dd2 22050es3 [1, -1, 1, -966755, 342138997] [2] 589824
22050.dd1 22050es4 [1, -1, 1, -2951255, -1950620003] [2] 589824

## Rank

sage: E.rank()

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

## Modular form 22050.2.a.dd

sage: E.q_eigenform(10)

$$q + q^{2} + q^{4} + q^{8} - 4q^{11} - 6q^{13} + q^{16} - 2q^{17} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.