# Properties

 Label 4641.d Number of curves $2$ Conductor $4641$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("d1")

sage: E.isogeny_class()

## Elliptic curves in class 4641.d

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
4641.d1 4641g1 $$[1, 0, 0, -455, 3696]$$ $$10418796526321/6390657$$ $$6390657$$ $$[2]$$ $$2240$$ $$0.24865$$ $$\Gamma_0(N)$$-optimal
4641.d2 4641g2 $$[1, 0, 0, -370, 5141]$$ $$-5602762882081/8312741073$$ $$-8312741073$$ $$[2]$$ $$4480$$ $$0.59522$$

## Rank

sage: E.rank()

The elliptic curves in class 4641.d have rank $$1$$.

## Complex multiplication

The elliptic curves in class 4641.d do not have complex multiplication.

## Modular form4641.2.a.d

sage: E.q_eigenform(10)

$$q - q^{2} + q^{3} - q^{4} - 4q^{5} - q^{6} + q^{7} + 3q^{8} + q^{9} + 4q^{10} - 4q^{11} - q^{12} + q^{13} - q^{14} - 4q^{15} - q^{16} + q^{17} - q^{18} + O(q^{20})$$

## 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.