# Properties

 Label 441525k Number of curves $6$ Conductor $441525$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("441525.k1")

sage: E.isogeny_class()

## Elliptic curves in class 441525k

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
441525.k5 441525k1 [1, 1, 1, -16484038, 27573593906]  41287680 $$\Gamma_0(N)$$-optimal*
441525.k4 441525k2 [1, 1, 1, -268889163, 1696981090656] [2, 2] 82575360 $$\Gamma_0(N)$$-optimal*
441525.k1 441525k3 [1, 1, 1, -4302220038, 108612515925156]  165150720 $$\Gamma_0(N)$$-optimal*
441525.k3 441525k4 [1, 1, 1, -274040288, 1628574150656] [2, 2] 165150720
441525.k6 441525k5 [1, 1, 1, 262412587, 7228069259906]  330301440
441525.k2 441525k6 [1, 1, 1, -892911163, -8348862096094]  330301440
*optimality has not been proved rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 3 curves highlighted, and conditionally curve 441525k1.

## Rank

sage: E.rank()

The elliptic curves in class 441525k have rank $$1$$.

## Modular form 441525.2.a.k

sage: E.q_eigenform(10)

$$q - q^{2} - q^{3} - q^{4} + q^{6} - q^{7} + 3q^{8} + q^{9} - 4q^{11} + q^{12} + 2q^{13} + q^{14} - q^{16} + 2q^{17} - q^{18} + 4q^{19} + O(q^{20})$$

## 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 Cremona numbering.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels. 