# Properties

 Label 14586.k Number of curves $2$ Conductor $14586$ CM no Rank $1$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 14586.k

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
14586.k1 14586j1 $$[1, 1, 1, -84338, -9461761]$$ $$66342819962001390625/4812668669952$$ $$4812668669952$$ $$$$ $$59136$$ $$1.4847$$ $$\Gamma_0(N)$$-optimal
14586.k2 14586j2 $$[1, 1, 1, -78898, -10728193]$$ $$-54315282059491182625/17983956399469632$$ $$-17983956399469632$$ $$$$ $$118272$$ $$1.8312$$

## Rank

sage: E.rank()

The elliptic curves in class 14586.k have rank $$1$$.

## Complex multiplication

The elliptic curves in class 14586.k do not have complex multiplication.

## Modular form 14586.2.a.k

sage: E.q_eigenform(10)

$$q + q^{2} - q^{3} + q^{4} - q^{6} - 4 q^{7} + q^{8} + q^{9} + q^{11} - q^{12} - q^{13} - 4 q^{14} + q^{16} - q^{17} + q^{18} + 2 q^{19} + O(q^{20})$$ ## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 