# Properties

 Label 6720.a Number of curves $4$ Conductor $6720$ CM no Rank $2$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("a1")

sage: E.isogeny_class()

## Elliptic curves in class 6720.a

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
6720.a1 6720bj3 $$[0, -1, 0, -4481, 116961]$$ $$303735479048/105$$ $$3440640$$ $$$$ $$6144$$ $$0.61016$$
6720.a2 6720bj2 $$[0, -1, 0, -281, 1881]$$ $$601211584/11025$$ $$45158400$$ $$[2, 2]$$ $$3072$$ $$0.26359$$
6720.a3 6720bj1 $$[0, -1, 0, -36, -30]$$ $$82881856/36015$$ $$2304960$$ $$$$ $$1536$$ $$-0.082988$$ $$\Gamma_0(N)$$-optimal
6720.a4 6720bj4 $$[0, -1, 0, -1, 5185]$$ $$-8/354375$$ $$-11612160000$$ $$$$ $$6144$$ $$0.61016$$

## Rank

sage: E.rank()

The elliptic curves in class 6720.a have rank $$2$$.

## Complex multiplication

The elliptic curves in class 6720.a do not have complex multiplication.

## Modular form6720.2.a.a

sage: E.q_eigenform(10)

$$q - q^{3} - q^{5} - q^{7} + q^{9} - 4 q^{11} - 6 q^{13} + q^{15} - 6 q^{17} - 4 q^{19} + O(q^{20})$$

## 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 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. 