# Properties

 Label 1850.f Number of curves $4$ Conductor $1850$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("f1")

sage: E.isogeny_class()

## Elliptic curves in class 1850.f

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1850.f1 1850a3 $$[1, 1, 0, -131875, -18487875]$$ $$16232905099479601/4052240$$ $$63316250000$$ $$$$ $$6912$$ $$1.4482$$
1850.f2 1850a4 $$[1, 1, 0, -131375, -18634375]$$ $$-16048965315233521/256572640900$$ $$-4008947514062500$$ $$$$ $$13824$$ $$1.7948$$
1850.f3 1850a1 $$[1, 1, 0, -1875, -17875]$$ $$46694890801/18944000$$ $$296000000000$$ $$$$ $$2304$$ $$0.89892$$ $$\Gamma_0(N)$$-optimal
1850.f4 1850a2 $$[1, 1, 0, 6125, -121875]$$ $$1625964918479/1369000000$$ $$-21390625000000$$ $$$$ $$4608$$ $$1.2455$$

## Rank

sage: E.rank()

The elliptic curves in class 1850.f have rank $$1$$.

## Complex multiplication

The elliptic curves in class 1850.f do not have complex multiplication.

## Modular form1850.2.a.f

sage: E.q_eigenform(10)

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

## 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 LMFDB numbering.

$$\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 