# Properties

 Label 441.e Number of curves $2$ Conductor $441$ CM $$\Q(\sqrt{-3})$$ Rank $0$ Graph # Related objects

Show commands: SageMath
E = EllipticCurve("e1")

E.isogeny_class()

## Elliptic curves in class 441.e

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality CM discriminant
441.e1 441a1 $$[0, 0, 1, 0, -4202]$$ $$0$$ $$-7626831723$$ $$[]$$ $$168$$ $$0.57513$$ $$\Gamma_0(N)$$-optimal $$-3$$
441.e2 441a2 $$[0, 0, 1, 0, 113447]$$ $$0$$ $$-5559960326067$$ $$[]$$ $$504$$ $$1.1244$$   $$-3$$

## Rank

sage: E.rank()

The elliptic curves in class 441.e have rank $$0$$.

## Complex multiplication

Each elliptic curve in class 441.e has complex multiplication by an order in the imaginary quadratic field $$\Q(\sqrt{-3})$$.

## Modular form441.2.a.e

sage: E.q_eigenform(10)

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

## 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 LMFDB numbering.

$$\left(\begin{array}{rr} 1 & 3 \\ 3 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 