# Properties

 Label 259920.ff Number of curves $4$ Conductor $259920$ CM no Rank $1$ Graph

# Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 259920.ff

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
259920.ff1 259920ff3 $$[0, 0, 0, -32283147, 70498544186]$$ $$26487576322129/44531250$$ $$6255671978217600000000$$ $$[4]$$ $$17694720$$ $$3.0776$$
259920.ff2 259920ff2 $$[0, 0, 0, -2652267, 350398874]$$ $$14688124849/8122500$$ $$1141034568826890240000$$ $$[2, 2]$$ $$8847360$$ $$2.7310$$
259920.ff3 259920ff1 $$[0, 0, 0, -1612587, -783476134]$$ $$3301293169/22800$$ $$3202904052847411200$$ $$[2]$$ $$4423680$$ $$2.3844$$ $$\Gamma_0(N)$$-optimal
259920.ff4 259920ff4 $$[0, 0, 0, 10343733, 2770254074]$$ $$871257511151/527800050$$ $$-74144426282371327795200$$ $$[2]$$ $$17694720$$ $$3.0776$$

## Rank

sage: E.rank()

The elliptic curves in class 259920.ff have rank $$1$$.

## Complex multiplication

The elliptic curves in class 259920.ff do not have complex multiplication.

## Modular form 259920.2.a.ff

sage: E.q_eigenform(10)

$$q + q^{5} + 4q^{11} - 2q^{13} - 2q^{17} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.