# Properties

 Label 169050gs Number of curves $6$ Conductor $169050$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("169050.ed1")

sage: E.isogeny_class()

## Elliptic curves in class 169050gs

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
169050.ed5 169050gs1 [1, 0, 1, -514526, -155893552] [2] 3538944 $$\Gamma_0(N)$$-optimal
169050.ed4 169050gs2 [1, 0, 1, -8452526, -9459229552] [2, 2] 7077888
169050.ed3 169050gs3 [1, 0, 1, -8673026, -8939731552] [2, 2] 14155776
169050.ed1 169050gs4 [1, 0, 1, -135240026, -605360479552] [2] 14155776
169050.ed2 169050gs5 [1, 0, 1, -31641776, 58680268448] [2] 28311552
169050.ed6 169050gs6 [1, 0, 1, 10767724, -43310977552] [2] 28311552

## Rank

sage: E.rank()

The elliptic curves in class 169050gs have rank $$0$$.

## Modular form 169050.2.a.ed

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.