# Properties

 Label 177870.ie Number of curves $4$ Conductor $177870$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("ie1")

sage: E.isogeny_class()

## Elliptic curves in class 177870.ie

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
177870.ie1 177870j3 [1, 0, 0, -2214605, 1268320395] [2] 3932160
177870.ie2 177870j2 [1, 0, 0, -139455, 19495125] [2, 2] 1966080
177870.ie3 177870j1 [1, 0, 0, -20875, -734623] [2] 983040 $$\Gamma_0(N)$$-optimal
177870.ie4 177870j4 [1, 0, 0, 38415, 65848047] [2] 3932160

## Rank

sage: E.rank()

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

## Complex multiplication

The elliptic curves in class 177870.ie do not have complex multiplication.

## Modular form 177870.2.a.ie

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.