# Properties

 Label 10780.k Number of curves $4$ Conductor $10780$ CM no Rank $1$ Graph # Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 10780.k

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
10780.k1 10780f4 [0, -1, 0, -347916, 79103816]  62208
10780.k2 10780f3 [0, -1, 0, -21821, 1232330]  31104
10780.k3 10780f2 [0, -1, 0, -4916, 76616]  20736
10780.k4 10780f1 [0, -1, 0, -2221, -38730]  10368 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 10780.k have rank $$1$$.

## Complex multiplication

The elliptic curves in class 10780.k do not have complex multiplication.

## Modular form 10780.2.a.k

sage: E.q_eigenform(10)

$$q + 2q^{3} - q^{5} + q^{9} - q^{11} + 4q^{13} - 2q^{15} + 4q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrr} 1 & 2 & 3 & 6 \\ 2 & 1 & 6 & 3 \\ 3 & 6 & 1 & 2 \\ 6 & 3 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 