# Properties

 Label 43681k Number of curves $2$ Conductor $43681$ CM no Rank $1$ Graph

# Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 43681k

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
43681.j2 43681k1 [1, 0, 1, -10838, 433365] [] 39312 $$\Gamma_0(N)$$-optimal
43681.j1 43681k2 [1, 0, 1, -110113, -54985911] [] 432432

## Rank

sage: E.rank()

The elliptic curves in class 43681k have rank $$1$$.

## Complex multiplication

The elliptic curves in class 43681k do not have complex multiplication.

## Modular form 43681.2.a.k

sage: E.q_eigenform(10)

$$q + q^{2} - 2q^{3} - q^{4} + q^{5} - 2q^{6} + 2q^{7} - 3q^{8} + q^{9} + q^{10} + 2q^{12} + q^{13} + 2q^{14} - 2q^{15} - q^{16} + 5q^{17} + q^{18} + O(q^{20})$$

## 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 Cremona numbering.

$$\left(\begin{array}{rr} 1 & 11 \\ 11 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels.