# Properties

 Label 4410.e Number of curves $2$ Conductor $4410$ CM no Rank $1$ Graph

# Related objects

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

sage: E.isogeny_class()

## Elliptic curves in class 4410.e

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
4410.e1 4410j1 $$[1, -1, 0, -135, 265]$$ $$1092727/540$$ $$135025380$$ $$[2]$$ $$1536$$ $$0.25317$$ $$\Gamma_0(N)$$-optimal
4410.e2 4410j2 $$[1, -1, 0, 495, 1651]$$ $$53582633/36450$$ $$-9114213150$$ $$[2]$$ $$3072$$ $$0.59975$$

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form4410.2.a.e

sage: E.q_eigenform(10)

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

## 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.