# Properties

 Label 840f Number of curves 6 Conductor 840 CM no Rank 1 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("840.d1")

sage: E.isogeny_class()

## Elliptic curves in class 840f

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
840.d4 840f1 [0, -1, 0, -175, 952] [4] 128 $$\Gamma_0(N)$$-optimal
840.d3 840f2 [0, -1, 0, -180, 900] [2, 4] 256
840.d2 840f3 [0, -1, 0, -680, -5700] [2, 2] 512
840.d5 840f4 [0, -1, 0, 240, 4092] [4] 512
840.d1 840f5 [0, -1, 0, -10480, -409460] [2] 1024
840.d6 840f6 [0, -1, 0, 1120, -32340] [2] 1024

## Rank

sage: E.rank()

The elliptic curves in class 840f have rank $$1$$.

## Modular form840.2.a.d

sage: E.q_eigenform(10)

$$q - q^{3} + q^{5} - q^{7} + q^{9} - 4q^{11} - 2q^{13} - q^{15} + 2q^{17} - 4q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.