# Properties

 Label 3850.k Number of curves $4$ Conductor $3850$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("k1")

sage: E.isogeny_class()

## Elliptic curves in class 3850.k

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
3850.k1 3850d4 $$[1, 1, 0, -652900, -197575500]$$ $$1969902499564819009/63690429687500$$ $$995162963867187500$$ $$$$ $$82944$$ $$2.2274$$
3850.k2 3850d2 $$[1, 1, 0, -89400, 10160000]$$ $$5057359576472449/51765560000$$ $$808836875000000$$ $$$$ $$27648$$ $$1.6781$$
3850.k3 3850d1 $$[1, 1, 0, -1400, 392000]$$ $$-19443408769/4249907200$$ $$-66404800000000$$ $$$$ $$13824$$ $$1.3315$$ $$\Gamma_0(N)$$-optimal
3850.k4 3850d3 $$[1, 1, 0, 12600, -10570000]$$ $$14156681599871/3100231750000$$ $$-48441121093750000$$ $$$$ $$41472$$ $$1.8808$$

## Rank

sage: E.rank()

The elliptic curves in class 3850.k have rank $$1$$.

## Complex multiplication

The elliptic curves in class 3850.k do not have complex multiplication.

## Modular form3850.2.a.k

sage: E.q_eigenform(10)

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