# Properties

 Label 47040cf Number of curves $8$ Conductor $47040$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("47040.ew1")

sage: E.isogeny_class()

## Elliptic curves in class 47040cf

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
47040.ew7 47040cf1 [0, 1, 0, -128641, 6205919] [2] 442368 $$\Gamma_0(N)$$-optimal
47040.ew5 47040cf2 [0, 1, 0, -1132161, -459628065] [2, 2] 884736
47040.ew4 47040cf3 [0, 1, 0, -8407681, 9380638175] [2] 1327104
47040.ew6 47040cf4 [0, 1, 0, -254081, -1153486881] [2] 1769472
47040.ew2 47040cf5 [0, 1, 0, -18066561, -29563087905] [2] 1769472
47040.ew3 47040cf6 [0, 1, 0, -8470401, 9233509599] [2, 2] 2654208
47040.ew8 47040cf7 [0, 1, 0, 2286079, 31092828255] [2] 5308416
47040.ew1 47040cf8 [0, 1, 0, -20230401, -22041034401] [2] 5308416

## Rank

sage: E.rank()

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

## Modular form 47040.2.a.ew

sage: E.q_eigenform(10)

$$q + q^{3} - q^{5} + q^{9} + 2q^{13} - q^{15} + 6q^{17} - 4q^{19} + O(q^{20})$$

## 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.