# Properties

 Label 177870.ff Number of curves 8 Conductor 177870 CM no Rank 0 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("177870.ff1")

sage: E.isogeny_class()

## Elliptic curves in class 177870.ff

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
177870.ff1 177870dy8 [1, 1, 1, -2082457616, -36578228819791] [2] 79626240
177870.ff2 177870dy6 [1, 1, 1, -130156496, -571548803407] [2, 2] 39813120
177870.ff3 177870dy7 [1, 1, 1, -120670096, -658383514447] [2] 79626240
177870.ff4 177870dy5 [1, 1, 1, -25835741, -49667231041] [2] 26542080
177870.ff5 177870dy3 [1, 1, 1, -8730576, -7549690191] [2] 19906560
177870.ff6 177870dy2 [1, 1, 1, -3424121, 1278863543] [2, 2] 13271040
177870.ff7 177870dy1 [1, 1, 1, -2949801, 1948034199] [2] 6635520 $$\Gamma_0(N)$$-optimal
177870.ff8 177870dy4 [1, 1, 1, 11398379, 9407522543] [2] 26542080

## Rank

sage: E.rank()

The elliptic curves in class 177870.ff have rank $$0$$.

## Modular form 177870.2.a.ff

sage: E.q_eigenform(10)

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

## 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.

$$\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 3 & 4 & 6 & 12 & 12 \\ 2 & 1 & 2 & 6 & 2 & 3 & 6 & 6 \\ 4 & 2 & 1 & 12 & 4 & 6 & 12 & 3 \\ 3 & 6 & 12 & 1 & 12 & 2 & 4 & 4 \\ 4 & 2 & 4 & 12 & 1 & 6 & 3 & 12 \\ 6 & 3 & 6 & 2 & 6 & 1 & 2 & 2 \\ 12 & 6 & 12 & 4 & 3 & 2 & 1 & 4 \\ 12 & 6 & 3 & 4 & 12 & 2 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels.