# Properties

 Label 441.f Number of curves 6 Conductor 441 CM no Rank 1 Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("441.f1")

sage: E.isogeny_class()

## Elliptic curves in class 441.f

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
441.f1 441c5 [1, -1, 0, -345753, -78165914]  1536
441.f2 441c3 [1, -1, 0, -21618, -1216265] [2, 2] 768
441.f3 441c4 [1, -1, 0, -17208, 867901]  768
441.f4 441c6 [1, -1, 0, -15003, -1979636]  1536
441.f5 441c2 [1, -1, 0, -1773, -5720] [2, 2] 384
441.f6 441c1 [1, -1, 0, 432, -869]  192 $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 441.f have rank $$1$$.

## Modular form441.2.a.f

sage: E.q_eigenform(10)

$$q + q^{2} - q^{4} - 2q^{5} - 3q^{8} - 2q^{10} - 4q^{11} + 2q^{13} - q^{16} - 6q^{17} - 4q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrrrr} 1 & 2 & 8 & 4 & 4 & 8 \\ 2 & 1 & 4 & 2 & 2 & 4 \\ 8 & 4 & 1 & 8 & 2 & 4 \\ 4 & 2 & 8 & 1 & 4 & 8 \\ 4 & 2 & 2 & 4 & 1 & 2 \\ 8 & 4 & 4 & 8 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 