# Properties

 Label 47610w Number of curves $6$ Conductor $47610$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("47610.x1")

sage: E.isogeny_class()

## Elliptic curves in class 47610w

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
47610.x5 47610w1 [1, -1, 0, -1999719, 1194630525] [2] 1622016 $$\Gamma_0(N)$$-optimal
47610.x4 47610w2 [1, -1, 0, -32850999, 72479598093] [2, 2] 3244032
47610.x3 47610w3 [1, -1, 0, -33707979, 68499268785] [2, 2] 6488064
47610.x1 47610w4 [1, -1, 0, -525614499, 4638327636393] [2] 6488064
47610.x6 47610w5 [1, -1, 0, 41849091, 331845880563] [2] 12976128
47610.x2 47610w6 [1, -1, 0, -122976729, -449598702465] [2] 12976128

## Rank

sage: E.rank()

The elliptic curves in class 47610w have rank $$0$$.

## Modular form 47610.2.a.x

sage: E.q_eigenform(10)

$$q - q^{2} + q^{4} + q^{5} - q^{8} - q^{10} + 4q^{11} - 2q^{13} + q^{16} - 6q^{17} - 4q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.