# Properties

 Label 281775bm Number of curves $6$ Conductor $281775$ CM no Rank $1$ Graph

# Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("281775.bm1")

sage: E.isogeny_class()

## Elliptic curves in class 281775bm

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
281775.bm4 281775bm1 [1, 1, 0, -3894425, 2956476000] [2] 4718592 $$\Gamma_0(N)$$-optimal
281775.bm3 281775bm2 [1, 1, 0, -3930550, 2898784375] [2, 2] 9437184
281775.bm5 281775bm3 [1, 1, 0, 1596575, 10410147250] [2] 18874368
281775.bm2 281775bm4 [1, 1, 0, -10035675, -8304120000] [2, 2] 18874368
281775.bm6 281775bm5 [1, 1, 0, 28003950, -56196007875] [2] 37748736
281775.bm1 281775bm6 [1, 1, 0, -145757300, -677276009625] [2] 37748736

## Rank

sage: E.rank()

The elliptic curves in class 281775bm have rank $$1$$.

## Modular form 281775.2.a.bm

sage: E.q_eigenform(10)

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

## 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.