# Properties

 Label 21840bk Number of curves 8 Conductor 21840 CM no Rank 0 Graph

# Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("21840.w1")

sage: E.isogeny_class()

## Elliptic curves in class 21840bk

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
21840.w7 21840bk1 [0, -1, 0, -411600, -100957248] [2] 221184 $$\Gamma_0(N)$$-optimal
21840.w6 21840bk2 [0, -1, 0, -662480, 36926400] [2, 2] 442368
21840.w5 21840bk3 [0, -1, 0, -2542800, 1493548992] [2] 663552
21840.w8 21840bk4 [0, -1, 0, 2603440, 290361792] [4] 884736
21840.w4 21840bk5 [0, -1, 0, -7942480, 8604030400] [2] 884736
21840.w2 21840bk6 [0, -1, 0, -40190480, 98082436800] [2, 2] 1327104
21840.w3 21840bk7 [0, -1, 0, -39696560, 100610121792] [4] 2654208
21840.w1 21840bk8 [0, -1, 0, -643047280, 6276641208640] [2] 2654208

## Rank

sage: E.rank()

The elliptic curves in class 21840bk have rank $$0$$.

## Modular form 21840.2.a.w

sage: E.q_eigenform(10)

$$q - q^{3} + q^{5} - q^{7} + q^{9} + q^{13} - q^{15} + 6q^{17} + 4q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels.