# Properties

 Label 84150.hc Number of curves $2$ Conductor $84150$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("hc1")

sage: E.isogeny_class()

## Elliptic curves in class 84150.hc

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
84150.hc1 84150fr1 $$[1, -1, 1, -203405, -34979403]$$ $$81706955619457/744505344$$ $$8480381184000000$$ $$[2]$$ $$1146880$$ $$1.8786$$ $$\Gamma_0(N)$$-optimal
84150.hc2 84150fr2 $$[1, -1, 1, -59405, -83651403]$$ $$-2035346265217/264305213568$$ $$-3010601573298000000$$ $$[2]$$ $$2293760$$ $$2.2252$$

## Rank

sage: E.rank()

The elliptic curves in class 84150.hc have rank $$0$$.

## Complex multiplication

The elliptic curves in class 84150.hc do not have complex multiplication.

## Modular form 84150.2.a.hc

sage: E.q_eigenform(10)

$$q + q^{2} + q^{4} + 4q^{7} + q^{8} + q^{11} + 4q^{13} + 4q^{14} + q^{16} - q^{17} - 8q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.