# Properties

 Label 47040.fw Number of curves $6$ Conductor $47040$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("47040.fw1")

sage: E.isogeny_class()

## Elliptic curves in class 47040.fw

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
47040.fw1 47040hf6 [0, 1, 0, -2054145, -1133828865] [2] 786432
47040.fw2 47040hf4 [0, 1, 0, -133345, -16307425] [2, 2] 393216
47040.fw3 47040hf2 [0, 1, 0, -35345, 2292975] [2, 2] 196608
47040.fw4 47040hf1 [0, 1, 0, -34365, 2440563] [2] 98304 $$\Gamma_0(N)$$-optimal
47040.fw5 47040hf3 [0, 1, 0, 46975, 11463423] [2] 393216
47040.fw6 47040hf5 [0, 1, 0, 219455, -87643585] [2] 786432

## Rank

sage: E.rank()

The elliptic curves in class 47040.fw have rank $$1$$.

## Modular form 47040.2.a.fw

sage: E.q_eigenform(10)

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

## 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 LMFDB numbering.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.