# Properties

 Label 169050.e Number of curves $6$ Conductor $169050$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("169050.e1")

sage: E.isogeny_class()

## Elliptic curves in class 169050.e

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
169050.e1 169050ia5 [1, 1, 0, -96939175, 367320339625] [2] 25165824
169050.e2 169050ia4 [1, 1, 0, -21699675, -38915017875] [2] 12582912
169050.e3 169050ia3 [1, 1, 0, -6215675, 5424298125] [2, 2] 12582912
169050.e4 169050ia2 [1, 1, 0, -1413675, -554191875] [2, 2] 6291456
169050.e5 169050ia1 [1, 1, 0, 154325, -47727875] [2] 3145728 $$\Gamma_0(N)$$-optimal
169050.e6 169050ia6 [1, 1, 0, 7675825, 26247656625] [2] 25165824

## Rank

sage: E.rank()

The elliptic curves in class 169050.e have rank $$1$$.

## Modular form 169050.2.a.e

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 LMFDB numbering.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.