# Properties

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

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("201810.bp1")

sage: E.isogeny_class()

## Elliptic curves in class 201810be

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
201810.bp7 201810be1 [1, 1, 1, -39421, 1043603] [2] 1451520 $$\Gamma_0(N)$$-optimal
201810.bp5 201810be2 [1, 1, 1, -346941, -78050541] [2, 2] 2903040
201810.bp4 201810be3 [1, 1, 1, -2576461, 1590706739] [2] 4354560
201810.bp6 201810be4 [1, 1, 1, -77861, -195692317] [2] 5806080
201810.bp2 201810be5 [1, 1, 1, -5536341, -5016283581] [2] 5806080
201810.bp3 201810be6 [1, 1, 1, -2595681, 1565743803] [2, 2] 8709120
201810.bp8 201810be7 [1, 1, 1, 700549, 5274661799] [2] 17418240
201810.bp1 201810be8 [1, 1, 1, -6199431, -3740417697] [2] 17418240

## Rank

sage: E.rank()

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

## Modular form 201810.2.a.bp

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.