Properties

Label 18810z
Number of curves 8
Conductor 18810
CM no
Rank 1
Graph

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Show commands for: SageMath

sage: E = EllipticCurve("18810.p1")
sage: E.isogeny_class()

Elliptic curves in class 18810z

sage: E.isogeny_class().curves
LMFDB label Cremona label Weierstrass coefficients Torsion order Modular degree Optimality
18810.p7 18810z1 [1, -1, 1, -778568, -231857269] 2 516096 \(\Gamma_0(N)\)-optimal
18810.p5 18810z2 [1, -1, 1, -12028568, -16053857269] 4 1032192  
18810.p4 18810z3 [1, -1, 1, -15712943, 23947250231] 6 1548288  
18810.p2 18810z4 [1, -1, 1, -192456068, -1027602593269] 2 2064384  
18810.p6 18810z5 [1, -1, 1, -11601068, -17248121269] 2 2064384  
18810.p3 18810z6 [1, -1, 1, -20320943, 8757439031] 12 3096576  
18810.p1 18810z7 [1, -1, 1, -192818543, -1023537198409] 6 6193152  
18810.p8 18810z8 [1, -1, 1, 78448657, 68848863671] 6 6193152  

Rank

sage: E.rank()

The elliptic curves in class 18810z have rank \(1\).

Modular form 18810.2.a.p

sage: E.q_eigenform(10)
\( q + q^{2} + q^{4} - q^{5} - 4q^{7} + q^{8} - q^{10} + q^{11} + 2q^{13} - 4q^{14} + q^{16} - 6q^{17} + q^{19} + O(q^{20}) \)

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.