# Properties

 Label 49686.bd Number of curves $6$ Conductor $49686$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("49686.bd1")

sage: E.isogeny_class()

## Elliptic curves in class 49686.bd

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
49686.bd1 49686bk4 [1, 0, 1, -11129837, -14292541924] [2] 1474560
49686.bd2 49686bk6 [1, 0, 1, -7569007, 7937471336] [2] 2949120
49686.bd3 49686bk3 [1, 0, 1, -861397, -108977620] [2, 2] 1474560
49686.bd4 49686bk2 [1, 0, 1, -695777, -223255420] [2, 2] 737280
49686.bd5 49686bk1 [1, 0, 1, -33297, -5167004] [2] 368640 $$\Gamma_0(N)$$-optimal
49686.bd6 49686bk5 [1, 0, 1, 3196293, -840984896] [2] 2949120

## Rank

sage: E.rank()

The elliptic curves in class 49686.bd have rank $$1$$.

## Modular form 49686.2.a.bd

sage: E.q_eigenform(10)

$$q - q^{2} + q^{3} + q^{4} - 2q^{5} - q^{6} - q^{8} + q^{9} + 2q^{10} + 4q^{11} + q^{12} - 2q^{15} + q^{16} - 2q^{17} - q^{18} - 4q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.