# Properties

 Label 6720.z Number of curves $4$ Conductor $6720$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("z1")

sage: E.isogeny_class()

## Elliptic curves in class 6720.z

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
6720.z1 6720bv3 [0, -1, 0, -23905, -1414655] [2] 12288
6720.z2 6720bv2 [0, -1, 0, -1505, -21375] [2, 2] 6144
6720.z3 6720bv1 [0, -1, 0, -225, 897] [2] 3072 $$\Gamma_0(N)$$-optimal
6720.z4 6720bv4 [0, -1, 0, 415, -73983] [4] 12288

## Rank

sage: E.rank()

The elliptic curves in class 6720.z have rank $$0$$.

## Complex multiplication

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

## Modular form6720.2.a.z

sage: E.q_eigenform(10)

$$q - q^{3} + q^{5} + q^{7} + q^{9} - 4q^{11} + 2q^{13} - q^{15} - 6q^{17} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.