# Properties

 Label 232050gk Number of curves 4 Conductor 232050 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("232050.gk1")

sage: E.isogeny_class()

## Elliptic curves in class 232050gk

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
232050.gk4 232050gk1 [1, 0, 0, -8313, -20101383]  2949120 $$\Gamma_0(N)$$-optimal
232050.gk3 232050gk2 [1, 0, 0, -1160313, -475141383] [2, 2] 5898240
232050.gk2 232050gk3 [1, 0, 0, -2252313, 561166617]  11796480
232050.gk1 232050gk4 [1, 0, 0, -18500313, -30629401383]  11796480

## Rank

sage: E.rank()

The elliptic curves in class 232050gk have rank $$1$$.

## Modular form 232050.2.a.gk

sage: E.q_eigenform(10)

$$q + q^{2} + q^{3} + q^{4} + q^{6} - q^{7} + q^{8} + q^{9} + q^{12} + q^{13} - q^{14} + q^{16} + q^{17} + q^{18} + 4q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels. 