# Properties

 Label 481650hr Number of curves $4$ Conductor $481650$ CM no Rank $0$ Graph

# Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 481650hr

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
481650.hr3 481650hr1 $$[1, 0, 0, -131063, 18142617]$$ $$3301293169/22800$$ $$1719550706250000$$ $$[2]$$ $$3538944$$ $$1.7570$$ $$\Gamma_0(N)$$-optimal
481650.hr2 481650hr2 $$[1, 0, 0, -215563, -8136883]$$ $$14688124849/8122500$$ $$612589939101562500$$ $$[2, 2]$$ $$7077888$$ $$2.1035$$
481650.hr4 481650hr3 $$[1, 0, 0, 840687, -64118133]$$ $$871257511151/527800050$$ $$-39806094242819531250$$ $$[2]$$ $$14155776$$ $$2.4501$$
481650.hr1 481650hr4 $$[1, 0, 0, -2623813, -1633705633]$$ $$26487576322129/44531250$$ $$3358497473144531250$$ $$[2]$$ $$14155776$$ $$2.4501$$

## Rank

sage: E.rank()

The elliptic curves in class 481650hr have rank $$0$$.

## Complex multiplication

The elliptic curves in class 481650hr do not have complex multiplication.

## Modular form 481650.2.a.hr

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.