# Properties

 Label 15870.bg Number of curves $6$ Conductor $15870$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("15870.bg1")

sage: E.isogeny_class()

## Elliptic curves in class 15870.bg

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
15870.bg1 15870bg3 [1, 0, 0, -58401611, -171789912459] [2] 811008
15870.bg2 15870bg5 [1, 0, 0, -13664081, 16651803795] [2] 1622016
15870.bg3 15870bg4 [1, 0, 0, -3745331, -2537009955] [2, 2] 811008
15870.bg4 15870bg2 [1, 0, 0, -3650111, -2684429559] [2, 2] 405504
15870.bg5 15870bg1 [1, 0, 0, -222191, -44245575] [4] 202752 $$\Gamma_0(N)$$-optimal
15870.bg6 15870bg6 [1, 0, 0, 4649899, -12290588169] [2] 1622016

## Rank

sage: E.rank()

The elliptic curves in class 15870.bg have rank $$0$$.

## Modular form 15870.2.a.bg

sage: E.q_eigenform(10)

$$q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + q^{8} + q^{9} - q^{10} - 4q^{11} + q^{12} - 2q^{13} - q^{15} + q^{16} + 6q^{17} + q^{18} - 4q^{19} + O(q^{20})$$

## 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.