# Properties

 Label 1850.e Number of curves $2$ Conductor $1850$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
E = EllipticCurve("e1")

E.isogeny_class()

## Elliptic curves in class 1850.e

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1850.e1 1850f2 $$[1, -1, 0, -13657, -610899]$$ $$2253707317528029/700928$$ $$87616000$$ $$$$ $$1728$$ $$0.88632$$
1850.e2 1850f1 $$[1, -1, 0, -857, -9299]$$ $$557238592989/9699328$$ $$1212416000$$ $$$$ $$864$$ $$0.53975$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form1850.2.a.e

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 