# Properties

 Label 43758.k Number of curves $2$ Conductor $43758$ CM no Rank $1$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 43758.k

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
43758.k1 43758q1 $$[1, -1, 1, -842, 6905]$$ $$90458382169/25788048$$ $$18799486992$$ $$$$ $$51200$$ $$0.67806$$ $$\Gamma_0(N)$$-optimal
43758.k2 43758q2 $$[1, -1, 1, 2218, 43625]$$ $$1656015369191/2114999172$$ $$-1541834396388$$ $$$$ $$102400$$ $$1.0246$$

## Rank

sage: E.rank()

The elliptic curves in class 43758.k have rank $$1$$.

## Complex multiplication

The elliptic curves in class 43758.k do not have complex multiplication.

## Modular form 43758.2.a.k

sage: E.q_eigenform(10)

$$q + q^{2} + q^{4} - 4q^{5} + q^{8} - 4q^{10} - q^{11} - q^{13} + q^{16} - q^{17} + 2q^{19} + O(q^{20})$$ ## 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. 