# Properties

 Label 139638.o Number of curves 6 Conductor 139638 CM no Rank 0 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("139638.o1")

sage: E.isogeny_class()

## Elliptic curves in class 139638.o

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
139638.o1 139638bd6 [1, 0, 1, -37981565, -90099554866] [2] 6635520
139638.o2 139638bd4 [1, 0, 1, -2373875, -1407920614] [2, 2] 3317760
139638.o3 139638bd5 [1, 0, 1, -2250665, -1560553162] [2] 6635520
139638.o4 139638bd2 [1, 0, 1, -156095, -19590334] [2, 2] 1658880
139638.o5 139638bd1 [1, 0, 1, -46575, 3584098] [2] 829440 $$\Gamma_0(N)$$-optimal
139638.o6 139638bd3 [1, 0, 1, 309365, -113985622] [2] 3317760

## Rank

sage: E.rank()

The elliptic curves in class 139638.o have rank $$0$$.

## Modular form 139638.2.a.o

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.