# Properties

 Label 205350v Number of curves $6$ Conductor $205350$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("205350.dx1")

sage: E.isogeny_class()

## Elliptic curves in class 205350v

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
205350.dx5 205350v1 [1, 0, 0, -28920838, 206074696292] [2] 56733696 $$\Gamma_0(N)$$-optimal
205350.dx4 205350v2 [1, 0, 0, -729848838, 7573528904292] [2, 2] 113467392
205350.dx1 205350v3 [1, 0, 0, -11670896838, 485292507728292] [2] 226934784
205350.dx3 205350v4 [1, 0, 0, -1003648838, 1374423104292] [2, 2] 226934784
205350.dx6 205350v5 [1, 0, 0, 3986356162, 10960222709292] [2] 453869568
205350.dx2 205350v6 [1, 0, 0, -10374453838, -404953052500708] [2] 453869568

## Rank

sage: E.rank()

The elliptic curves in class 205350v have rank $$1$$.

## Modular form 205350.2.a.dx

sage: E.q_eigenform(10)

$$q + q^{2} + q^{3} + q^{4} + q^{6} + q^{8} + q^{9} + 4q^{11} + q^{12} - 2q^{13} + q^{16} + 2q^{17} + q^{18} - 4q^{19} + O(q^{20})$$

## 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 Cremona numbering.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.