# Properties

 Label 22050.fn Number of curves $6$ Conductor $22050$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("22050.fn1")

sage: E.isogeny_class()

## Elliptic curves in class 22050.fn

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
22050.fn1 22050eq4 [1, -1, 1, -14817830, 21958278297] [2] 786432
22050.fn2 22050eq6 [1, -1, 1, -10077080, -12191879703] [2] 1572864
22050.fn3 22050eq3 [1, -1, 1, -1146830, 167586297] [2, 2] 786432
22050.fn4 22050eq2 [1, -1, 1, -926330, 343104297] [2, 2] 393216
22050.fn5 22050eq1 [1, -1, 1, -44330, 7944297] [2] 196608 $$\Gamma_0(N)$$-optimal
22050.fn6 22050eq5 [1, -1, 1, 4255420, 1291254297] [2] 1572864

## Rank

sage: E.rank()

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

## Modular form 22050.2.a.fn

sage: E.q_eigenform(10)

$$q + q^{2} + q^{4} + q^{8} + 4q^{11} + 6q^{13} + q^{16} - 2q^{17} + 4q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.