# Properties

 Label 441f Number of curves $2$ Conductor $441$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("f1")

sage: E.isogeny_class()

## Elliptic curves in class 441f

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
441.a2 441f1 $$[0, 0, 1, -21, 40]$$ $$-28672/3$$ $$-107163$$ $$[]$$ $$48$$ $$-0.29578$$ $$\Gamma_0(N)$$-optimal
441.a1 441f2 $$[0, 0, 1, -8211, -286610]$$ $$-1713910976512/1594323$$ $$-56950811883$$ $$[]$$ $$624$$ $$0.98669$$

## Rank

sage: E.rank()

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

## Complex multiplication

The elliptic curves in class 441f do not have complex multiplication.

## Modular form441.2.a.f

sage: E.q_eigenform(10)

$$q - 2q^{2} + 2q^{4} - 2q^{5} + 4q^{10} + 2q^{11} - q^{13} - 4q^{16} - q^{19} + O(q^{20})$$ ## 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.

$$\left(\begin{array}{rr} 1 & 13 \\ 13 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels. 