# Properties

 Label 1856.k Number of curves $2$ Conductor $1856$ CM no Rank $0$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 1856.k

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
1856.k1 1856a2 $$[0, 1, 0, -29121, -1935457]$$ $$-10418796526321/82044596$$ $$-21507498573824$$ $$[]$$ $$3840$$ $$1.3869$$
1856.k2 1856a1 $$[0, 1, 0, 319, 3743]$$ $$13651919/29696$$ $$-7784628224$$ $$[]$$ $$768$$ $$0.58217$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 1856.k have rank $$0$$.

## Complex multiplication

The elliptic curves in class 1856.k do not have complex multiplication.

## Modular form1856.2.a.k

sage: E.q_eigenform(10)

$$q + q^{3} - q^{5} - 2 q^{7} - 2 q^{9} + 3 q^{11} + q^{13} - q^{15} + 8 q^{17} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rr} 1 & 5 \\ 5 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 