# Properties

 Label 177450fb Number of curves 8 Conductor 177450 CM no Rank 0 Graph

# Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("177450.gh1")

sage: E.isogeny_class()

## Elliptic curves in class 177450fb

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
177450.gh7 177450fb1 [1, 1, 1, 887162, 242726531] [2] 7077888 $$\Gamma_0(N)$$-optimal
177450.gh6 177450fb2 [1, 1, 1, -4520838, 2167974531] [2, 2] 14155776
177450.gh4 177450fb3 [1, 1, 1, -63670838, 195470174531] [2] 28311552
177450.gh5 177450fb4 [1, 1, 1, -31898838, -67810193469] [2, 2] 28311552
177450.gh8 177450fb5 [1, 1, 1, 5365662, -216719135469] [2] 56623104
177450.gh2 177450fb6 [1, 1, 1, -507211338, -4396956443469] [2, 2] 56623104
177450.gh3 177450fb7 [1, 1, 1, -504042588, -4454602343469] [2] 113246208
177450.gh1 177450fb8 [1, 1, 1, -8115380088, -281395164293469] [2] 113246208

## Rank

sage: E.rank()

The elliptic curves in class 177450fb have rank $$0$$.

## Modular form 177450.2.a.gh

sage: E.q_eigenform(10)

$$q + q^{2} - q^{3} + q^{4} - q^{6} - q^{7} + q^{8} + q^{9} + 4q^{11} - q^{12} - q^{14} + q^{16} - 2q^{17} + q^{18} - 4q^{19} + O(q^{20})$$

## 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 Cremona numbering.

$$\left(\begin{array}{rrrrrrrr} 1 & 2 & 4 & 4 & 8 & 8 & 16 & 16 \\ 2 & 1 & 2 & 2 & 4 & 4 & 8 & 8 \\ 4 & 2 & 1 & 4 & 8 & 8 & 16 & 16 \\ 4 & 2 & 4 & 1 & 2 & 2 & 4 & 4 \\ 8 & 4 & 8 & 2 & 1 & 4 & 8 & 8 \\ 8 & 4 & 8 & 2 & 4 & 1 & 2 & 2 \\ 16 & 8 & 16 & 4 & 8 & 2 & 1 & 4 \\ 16 & 8 & 16 & 4 & 8 & 2 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels.