# Properties

 Label 106470.ff Number of curves $4$ Conductor $106470$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("106470.ff1")

sage: E.isogeny_class()

## Elliptic curves in class 106470.ff

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
106470.ff1 106470dn3 [1, -1, 1, -71012, -6364601] [2] 663552
106470.ff2 106470dn1 [1, -1, 1, -17777, 915581] [2] 221184 $$\Gamma_0(N)$$-optimal
106470.ff3 106470dn2 [1, -1, 1, -12707, 1444889] [2] 442368
106470.ff4 106470dn4 [1, -1, 1, 111508, -33815609] [2] 1327104

## Rank

sage: E.rank()

The elliptic curves in class 106470.ff have rank $$1$$.

## Modular form 106470.2.a.ff

sage: E.q_eigenform(10)

$$q + q^{2} + q^{4} + q^{5} - q^{7} + q^{8} + q^{10} - q^{14} + q^{16} - 2q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.