# Properties

 Label 177450hr Number of curves $4$ Conductor $177450$ CM no Rank $0$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("hr1")

sage: E.isogeny_class()

## Elliptic curves in class 177450hr

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
177450.do3 177450hr1 [1, 0, 1, -14876, -442102]  884736 $$\Gamma_0(N)$$-optimal
177450.do2 177450hr2 [1, 0, 1, -99376, 11725898] [2, 2] 1769472
177450.do1 177450hr3 [1, 0, 1, -1578126, 762930898]  3538944
177450.do4 177450hr4 [1, 0, 1, 27374, 39610898]  3538944

## Rank

sage: E.rank()

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

## Complex multiplication

The elliptic curves in class 177450hr do not have complex multiplication.

## Modular form 177450.2.a.hr

sage: E.q_eigenform(10)

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

## 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 Cremona 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 Cremona labels. 