# Properties

 Label 210c Number of curves 6 Conductor 210 CM no Rank 0 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("210.c1")

sage: E.isogeny_class()

## Elliptic curves in class 210c

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
210.c6 210c1 [1, 1, 1, 10, -13] [4] 32 $$\Gamma_0(N)$$-optimal
210.c5 210c2 [1, 1, 1, -70, -205] [2, 4] 64
210.c2 210c3 [1, 1, 1, -1050, -13533] [2, 2] 128
210.c4 210c4 [1, 1, 1, -370, 2435] [4] 128
210.c1 210c5 [1, 1, 1, -16800, -845133] [2] 256
210.c3 210c6 [1, 1, 1, -980, -15325] [2] 256

## Rank

sage: E.rank()

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

## Modular form210.2.a.c

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 Cremona numbering.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.