Properties

 Label 201810cg Number of curves 8 Conductor 201810 CM no Rank 0 Graph

Related objects

Show commands for: SageMath
sage: E = EllipticCurve("201810.p1")

sage: E.isogeny_class()

Elliptic curves in class 201810cg

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
201810.p7 201810cg1 [1, 1, 0, -478117, -127398131] [2] 2903040 $$\Gamma_0(N)$$-optimal
201810.p6 201810cg2 [1, 1, 0, -554997, -83776419] [2, 2] 5806080
201810.p5 201810cg3 [1, 1, 0, -1415092, 491827024] [2] 8709120
201810.p4 201810cg4 [1, 1, 0, -4187577, 3238581249] [2] 11612160
201810.p8 201810cg5 [1, 1, 0, 1847503, -612806919] [2] 11612160
201810.p2 201810cg6 [1, 1, 0, -21096372, 37284011856] [2, 2] 17418240
201810.p1 201810cg7 [1, 1, 0, -337534452, 2386710180624] [2] 34836480
201810.p3 201810cg8 [1, 1, 0, -19558772, 42951297936] [2] 34836480

Rank

sage: E.rank()

The elliptic curves in class 201810cg have rank $$0$$.

Modular form 201810.2.a.p

sage: E.q_eigenform(10)

$$q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} + q^{7} - q^{8} + q^{9} - q^{10} - q^{12} - 2q^{13} - q^{14} - q^{15} + q^{16} + 6q^{17} - q^{18} + 8q^{19} + O(q^{20})$$

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.

$$\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)$$

Isogeny graph

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

The vertices are labelled with Cremona labels.