# Properties

 Label 4641.c Number of curves $4$ Conductor $4641$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("c1")

sage: E.isogeny_class()

## Elliptic curves in class 4641.c

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
4641.c1 4641a3 $$[1, 1, 1, -24752, -1509184]$$ $$1677087406638588673/4641$$ $$4641$$ $$$$ $$5120$$ $$0.82194$$
4641.c2 4641a2 $$[1, 1, 1, -1547, -24064]$$ $$409460675852593/21538881$$ $$21538881$$ $$[2, 2]$$ $$2560$$ $$0.47537$$
4641.c3 4641a4 $$[1, 1, 1, -1462, -26716]$$ $$-345608484635233/94427721297$$ $$-94427721297$$ $$$$ $$5120$$ $$0.82194$$
4641.c4 4641a1 $$[1, 1, 1, -102, -366]$$ $$117433042273/22801233$$ $$22801233$$ $$$$ $$1280$$ $$0.12880$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

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

## Complex multiplication

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

## Modular form4641.2.a.c

sage: E.q_eigenform(10)

$$q - q^{2} - q^{3} - q^{4} + 2q^{5} + q^{6} - q^{7} + 3q^{8} + q^{9} - 2q^{10} + 4q^{11} + q^{12} + q^{13} + q^{14} - 2q^{15} - q^{16} + q^{17} - q^{18} + 4q^{19} + O(q^{20})$$ ## 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 LMFDB numbering.

$$\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 