Properties

 Label 159936fx Number of curves 6 Conductor 159936 CM no Rank 1 Graph

Related objects

Show commands for: SageMath
sage: E = EllipticCurve("159936.gx1")

sage: E.isogeny_class()

Elliptic curves in class 159936fx

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
159936.gx5 159936fx1 [0, 1, 0, -106689, -12475233] [2] 1179648 $$\Gamma_0(N)$$-optimal
159936.gx4 159936fx2 [0, 1, 0, -357569, 67756191] [2, 2] 2359296
159936.gx2 159936fx3 [0, 1, 0, -5437889, 4878819231] [2, 2] 4718592
159936.gx6 159936fx4 [0, 1, 0, 708671, 395518367] [2] 4718592
159936.gx1 159936fx5 [0, 1, 0, -87005249, 312338826015] [2] 9437184
159936.gx3 159936fx6 [0, 1, 0, -5155649, 5408132127] [2] 9437184

Rank

sage: E.rank()

The elliptic curves in class 159936fx have rank $$1$$.

Modular form 159936.2.a.gx

sage: E.q_eigenform(10)

$$q + q^{3} - 2q^{5} + q^{9} + 4q^{11} - 2q^{13} - 2q^{15} - q^{17} + 4q^{19} + O(q^{20})$$

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 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.