# Properties

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

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("22050.df1")

sage: E.isogeny_class()

## Elliptic curves in class 22050.df

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
22050.df1 22050er6 [1, -1, 1, -185220230, -970197193353] [2] 2359296
22050.df2 22050er4 [1, -1, 1, -11576480, -15156568353] [2, 2] 1179648
22050.df3 22050er5 [1, -1, 1, -10804730, -17264989353] [2] 2359296
22050.df4 22050er3 [1, -1, 1, -4079480, 2998519647] [2] 1179648
22050.df5 22050er2 [1, -1, 1, -771980, -203140353] [2, 2] 589824
22050.df6 22050er1 [1, -1, 1, 110020, -19684353] [2] 294912 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

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

## Modular form 22050.2.a.df

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.