# Properties

 Label 29040.df Number of curves 8 Conductor 29040 CM no Rank 1 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("29040.df1")

sage: E.isogeny_class()

## Elliptic curves in class 29040.df

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
29040.df1 29040dg8 [0, 1, 0, -4181800, -3292887820] [2] 327680
29040.df2 29040dg6 [0, 1, 0, -261400, -51501100] [2, 2] 163840
29040.df3 29040dg7 [0, 1, 0, -213000, -71112780] [2] 327680
29040.df4 29040dg4 [0, 1, 0, -154920, 23418228] [2] 81920
29040.df5 29040dg3 [0, 1, 0, -19400, -487500] [2, 2] 81920
29040.df6 29040dg2 [0, 1, 0, -9720, 360468] [2, 2] 40960
29040.df7 29040dg1 [0, 1, 0, -40, 15860] [2] 20480 $$\Gamma_0(N)$$-optimal
29040.df8 29040dg5 [0, 1, 0, 67720, -3588972] [2] 163840

## Rank

sage: E.rank()

The elliptic curves in class 29040.df have rank $$1$$.

## Modular form 29040.2.a.df

sage: E.q_eigenform(10)

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

## 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 LMFDB numbering.

$$\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 16 & 4 & 8 & 16 & 8 \\ 2 & 1 & 2 & 8 & 2 & 4 & 8 & 4 \\ 4 & 2 & 1 & 16 & 4 & 8 & 16 & 8 \\ 16 & 8 & 16 & 1 & 4 & 2 & 4 & 8 \\ 4 & 2 & 4 & 4 & 1 & 2 & 4 & 2 \\ 8 & 4 & 8 & 2 & 2 & 1 & 2 & 4 \\ 16 & 8 & 16 & 4 & 4 & 2 & 1 & 8 \\ 8 & 4 & 8 & 8 & 2 & 4 & 8 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.