# Properties

 Label 2850.q Number of curves $4$ Conductor $2850$ CM no Rank $1$ Graph

# Related objects

Show commands: SageMath
sage: E = EllipticCurve("q1")

sage: E.isogeny_class()

## Elliptic curves in class 2850.q

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
2850.q1 2850r4 $$[1, 1, 1, -186938, -30118969]$$ $$46237740924063961/1806561830400$$ $$28227528600000000$$ $$[2]$$ $$20736$$ $$1.9238$$
2850.q2 2850r2 $$[1, 1, 1, -27563, 1737281]$$ $$148212258825961/1218375000$$ $$19037109375000$$ $$[2]$$ $$6912$$ $$1.3745$$
2850.q3 2850r1 $$[1, 1, 1, -563, 63281]$$ $$-1263214441/110808000$$ $$-1731375000000$$ $$[2]$$ $$3456$$ $$1.0279$$ $$\Gamma_0(N)$$-optimal
2850.q4 2850r3 $$[1, 1, 1, 5062, -1702969]$$ $$918046641959/80912056320$$ $$-1264250880000000$$ $$[2]$$ $$10368$$ $$1.5772$$

## Rank

sage: E.rank()

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

## Complex multiplication

The elliptic curves in class 2850.q do not have complex multiplication.

## Modular form2850.2.a.q

sage: E.q_eigenform(10)

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

## 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 LMFDB numbering.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.