# Properties

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

# Related objects

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 structure 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] [2, 2] 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] [2, 6] 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.