# Properties

 Label 210b Number of curves 8 Conductor 210 CM no Rank 0 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("210.b1")

sage: E.isogeny_class()

## Elliptic curves in class 210b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
210.b7 210b1 [1, 0, 1, -498, 4228] [6] 96 $$\Gamma_0(N)$$-optimal
210.b6 210b2 [1, 0, 1, -578, 2756] [2, 6] 192
210.b5 210b3 [1, 0, 1, -1473, -16652] [2] 288
210.b4 210b4 [1, 0, 1, -4358, -109132] [6] 384
210.b8 210b5 [1, 0, 1, 1922, 20756] [12] 384
210.b2 210b6 [1, 0, 1, -21953, -1253644] [2, 2] 576
210.b1 210b7 [1, 0, 1, -351233, -80149132] [2] 1152
210.b3 210b8 [1, 0, 1, -20353, -1443724] [4] 1152

## Rank

sage: E.rank()

The elliptic curves in class 210b have rank $$0$$.

## Modular form210.2.a.b

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.