# Properties

 Label 6720bg Number of curves $6$ Conductor $6720$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("6720.k1")

sage: E.isogeny_class()

## Elliptic curves in class 6720bg

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
6720.k6 6720bg1 [0, -1, 0, 639, 8481] [2] 6144 $$\Gamma_0(N)$$-optimal
6720.k5 6720bg2 [0, -1, 0, -4481, 91425] [2, 2] 12288
6720.k4 6720bg3 [0, -1, 0, -23681, -1317855] [2] 24576
6720.k2 6720bg4 [0, -1, 0, -67201, 6727201] [2, 2] 24576
6720.k1 6720bg5 [0, -1, 0, -1075201, 429482401] [2] 49152
6720.k3 6720bg6 [0, -1, 0, -62721, 7658145] [2] 49152

## Rank

sage: E.rank()

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

## Modular form6720.2.a.k

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 Cremona numbering.

$$\left(\begin{array}{rrrrrr} 1 & 2 & 4 & 4 & 8 & 8 \\ 2 & 1 & 2 & 2 & 4 & 4 \\ 4 & 2 & 1 & 4 & 8 & 8 \\ 4 & 2 & 4 & 1 & 2 & 2 \\ 8 & 4 & 8 & 2 & 1 & 4 \\ 8 & 4 & 8 & 2 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels.