# Properties

 Label 27225bm Number of curves $2$ Conductor $27225$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("bm1")

sage: E.isogeny_class()

## Elliptic curves in class 27225bm

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
27225.g2 27225bm1 [1, -1, 1, -6755, 215372] [] 20160 $$\Gamma_0(N)$$-optimal
27225.g1 27225bm2 [1, -1, 1, -68630, -27034378] [] 221760

## Rank

sage: E.rank()

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

## Complex multiplication

The elliptic curves in class 27225bm do not have complex multiplication.

## Modular form 27225.2.a.bm

sage: E.q_eigenform(10)

$$q - q^{2} - q^{4} - 2q^{7} + 3q^{8} + q^{13} + 2q^{14} - q^{16} + 5q^{17} - 6q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rr} 1 & 11 \\ 11 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels. 