# Properties

 Label 227850.cy Number of curves $6$ Conductor $227850$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("227850.cy1")

sage: E.isogeny_class()

## Elliptic curves in class 227850.cy

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
227850.cy1 227850ga5 [1, 0, 1, -376712026, 2814216436448] [2] 37748736
227850.cy2 227850ga3 [1, 0, 1, -23544526, 43970566448] [2, 2] 18874368
227850.cy3 227850ga6 [1, 0, 1, -23177026, 45409696448] [2] 37748736
227850.cy4 227850ga4 [1, 0, 1, -4532526, -2896169552] [2] 18874368
227850.cy5 227850ga2 [1, 0, 1, -1494526, 664366448] [2, 2] 9437184
227850.cy6 227850ga1 [1, 0, 1, 73474, 43438448] [2] 4718592 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 227850.cy have rank $$0$$.

## Modular form 227850.2.a.cy

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.