# Properties

 Label 441d Number of curves 4 Conductor 441 CM -7 Rank 1 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("441.c1")

sage: E.isogeny_class()

## Elliptic curves in class 441d

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
441.c4 441d1 [1, -1, 1, -20, 46] [2] 32 $$\Gamma_0(N)$$-optimal
441.c3 441d2 [1, -1, 1, -335, 2440] [2] 64
441.c2 441d3 [1, -1, 1, -965, -13940] [2] 224
441.c1 441d4 [1, -1, 1, -16400, -804212] [2] 448

## Rank

sage: E.rank()

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

## Modular form441.2.a.c

sage: E.q_eigenform(10)

$$q - q^{2} - q^{4} + 3q^{8} - 4q^{11} - q^{16} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrr} 1 & 2 & 7 & 14 \\ 2 & 1 & 14 & 7 \\ 7 & 14 & 1 & 2 \\ 14 & 7 & 2 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels.