# Properties

 Label 210.a Number of curves 4 Conductor 210 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("210.a1")

sage: E.isogeny_class()

## Elliptic curves in class 210.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
210.a1 210d3 [1, 1, 0, -373, 2623]  64
210.a2 210d2 [1, 1, 0, -23, 33] [2, 2] 32
210.a3 210d1 [1, 1, 0, -3, -3]  16 $$\Gamma_0(N)$$-optimal
210.a4 210d4 [1, 1, 0, 7, 147]  64

## Rank

sage: E.rank()

The elliptic curves in class 210.a have rank $$1$$.

## Modular form210.2.a.a

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 