# Properties

 Label 4410k Number of curves $2$ Conductor $4410$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("k1")

sage: E.isogeny_class()

## Elliptic curves in class 4410k

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
4410.c2 4410k1 $$[1, -1, 0, -450, -138020]$$ $$-49/40$$ $$-8236978260840$$ $$[]$$ $$10080$$ $$1.1573$$ $$\Gamma_0(N)$$-optimal
4410.c1 4410k2 $$[1, -1, 0, -216540, -38731694]$$ $$-5452947409/250$$ $$-51481114130250$$ $$[]$$ $$30240$$ $$1.7066$$

## Rank

sage: E.rank()

The elliptic curves in class 4410k have rank $$1$$.

## Complex multiplication

The elliptic curves in class 4410k do not have complex multiplication.

## Modular form4410.2.a.k

sage: E.q_eigenform(10)

$$q - q^{2} + q^{4} - q^{5} - q^{8} + q^{10} - 3q^{11} - 5q^{13} + q^{16} + 6q^{17} + q^{19} + O(q^{20})$$ ## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels. 