# Properties

 Label 6720.bz Number of curves $4$ Conductor $6720$ CM no Rank $1$ Graph # Related objects

Show commands: SageMath
sage: E = EllipticCurve("bz1")

sage: E.isogeny_class()

## Elliptic curves in class 6720.bz

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
6720.bz1 6720ba4 $$[0, 1, 0, -1296065, 567489663]$$ $$7347751505995469192/72930375$$ $$2389782528000$$ $$$$ $$61440$$ $$1.9515$$
6720.bz2 6720ba3 $$[0, 1, 0, -116065, 425663]$$ $$5276930158229192/3050936350875$$ $$99973082345472000$$ $$$$ $$61440$$ $$1.9515$$
6720.bz3 6720ba2 $$[0, 1, 0, -81065, 8832663]$$ $$14383655824793536/45209390625$$ $$185177664000000$$ $$[2, 2]$$ $$30720$$ $$1.6049$$
6720.bz4 6720ba1 $$[0, 1, 0, -2940, 254538]$$ $$-43927191786304/415283203125$$ $$-26578125000000$$ $$$$ $$15360$$ $$1.2583$$ $$\Gamma_0(N)$$-optimal

## Rank

sage: E.rank()

The elliptic curves in class 6720.bz have rank $$1$$.

## Complex multiplication

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

## Modular form6720.2.a.bz

sage: E.q_eigenform(10)

$$q + q^{3} + q^{5} - q^{7} + q^{9} - 4q^{11} + 2q^{13} + q^{15} - 2q^{17} + 4q^{19} + O(q^{20})$$ ## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 