# Properties

 Label 2850b Number of curves $4$ Conductor $2850$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("2850.a1")

sage: E.isogeny_class()

## Elliptic curves in class 2850b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
2850.a4 2850b1 [1, 1, 0, -250, 2500]  2304 $$\Gamma_0(N)$$-optimal
2850.a3 2850b2 [1, 1, 0, -4750, 124000] [2, 2] 4608
2850.a2 2850b3 [1, 1, 0, -5500, 81250]  9216
2850.a1 2850b4 [1, 1, 0, -76000, 8032750]  9216

## Rank

sage: E.rank()

The elliptic curves in class 2850b have rank $$1$$.

## Modular form2850.2.a.a

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels. 