# Properties

 Label 441b Number of curves $2$ Conductor $441$ CM $$\Q(\sqrt{-3})$$ Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("b1")

sage: E.isogeny_class()

## Elliptic curves in class 441b

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality CM discriminant
441.d2 441b1 $$[0, 0, 1, 0, 12]$$ $$0$$ $$-64827$$ $$$$ $$24$$ $$-0.39783$$ $$\Gamma_0(N)$$-optimal $$-3$$
441.d1 441b2 $$[0, 0, 1, 0, -331]$$ $$0$$ $$-47258883$$ $$[]$$ $$72$$ $$0.15148$$   $$-3$$

## Rank

sage: E.rank()

The elliptic curves in class 441b have rank $$1$$.

## Complex multiplication

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

## Modular form441.2.a.b

sage: E.q_eigenform(10)

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

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

## Isogeny graph

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

The vertices are labelled with Cremona labels. 