# Properties

 Label 15870u Number of curves 8 Conductor 15870 CM no Rank 0 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("15870.u1")

sage: E.isogeny_class()

## Elliptic curves in class 15870u

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
15870.u8 15870u1 [1, 0, 1, 782, -25804] [2] 25344 $$\Gamma_0(N)$$-optimal
15870.u6 15870u2 [1, 0, 1, -9798, -338972] [2, 2] 50688
15870.u7 15870u3 [1, 0, 1, -7153, 761348] [2] 76032
15870.u4 15870u4 [1, 0, 1, -152628, -22963244] [2] 101376
15870.u5 15870u5 [1, 0, 1, -36248, 2284868] [2] 101376
15870.u3 15870u6 [1, 0, 1, -176433, 28455556] [2, 2] 152064
15870.u2 15870u7 [1, 0, 1, -239913, 6135988] [2] 304128
15870.u1 15870u8 [1, 0, 1, -2821433, 1823881556] [2] 304128

## Rank

sage: E.rank()

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

## Modular form 15870.2.a.u

sage: E.q_eigenform(10)

$$q - q^{2} + q^{3} + q^{4} + q^{5} - q^{6} + 4q^{7} - q^{8} + q^{9} - q^{10} + q^{12} + 2q^{13} - 4q^{14} + q^{15} + q^{16} - 6q^{17} - q^{18} + 4q^{19} + O(q^{20})$$

## 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 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.