# Properties

 Label 27225.r Number of curves 4 Conductor 27225 CM no Rank 1 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("27225.r1")

sage: E.isogeny_class()

## Elliptic curves in class 27225.r

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
27225.r1 27225bo4 [1, -1, 1, -3989030, 3064527722] [2] 737280
27225.r2 27225bo2 [1, -1, 1, -313655, 21317222] [2, 2] 368640
27225.r3 27225bo1 [1, -1, 1, -177530, -28504528] [2] 184320 $$\Gamma_0(N)$$-optimal
27225.r4 27225bo3 [1, -1, 1, 1183720, 165065222] [2] 737280

## Rank

sage: E.rank()

The elliptic curves in class 27225.r have rank $$1$$.

## Modular form 27225.2.a.r

sage: E.q_eigenform(10)

$$q - q^{2} - q^{4} + 4q^{7} + 3q^{8} - 2q^{13} - 4q^{14} - q^{16} + 2q^{17} + O(q^{20})$$

## 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.