# Properties

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

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("210.e1")

sage: E.isogeny_class()

## Elliptic curves in class 210.e

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
210.e1 210e7 [1, 0, 0, -1920800, -1024800150] [2] 2048
210.e2 210e5 [1, 0, 0, -120050, -16020000] [2, 2] 1024
210.e3 210e8 [1, 0, 0, -119300, -16229850] [2] 2048
210.e4 210e4 [1, 0, 0, -15070, 710612] [8] 512
210.e5 210e3 [1, 0, 0, -7550, -247500] [2, 4] 512
210.e6 210e2 [1, 0, 0, -1070, 7812] [2, 8] 256
210.e7 210e1 [1, 0, 0, 210, 900] [8] 128 $$\Gamma_0(N)$$-optimal
210.e8 210e6 [1, 0, 0, 1270, -789048] [4] 1024

## Rank

sage: E.rank()

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

## Modular form210.2.a.e

sage: E.q_eigenform(10)

$$q + q^{2} + q^{3} + q^{4} + q^{5} + q^{6} - q^{7} + q^{8} + q^{9} + q^{10} - 4q^{11} + q^{12} - 2q^{13} - q^{14} + q^{15} + q^{16} + 2q^{17} + q^{18} + 4q^{19} + O(q^{20})$$

## 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 LMFDB numbering.

$$\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 16 & 4 & 8 & 16 & 8 \\ 2 & 1 & 2 & 8 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 16 & 4 & 8 & 16 & 8 \\ 16 & 8 & 16 & 1 & 4 & 2 & 4 & 8 \\ 4 & 2 & 4 & 4 & 1 & 2 & 4 & 2 \\ 8 & 4 & 8 & 2 & 2 & 1 & 2 & 4 \\ 16 & 8 & 16 & 4 & 4 & 2 & 1 & 8 \\ 8 & 4 & 8 & 8 & 2 & 4 & 8 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.