# Properties

 Label 6720.m Number of curves $6$ Conductor $6720$ CM no Rank $1$ Graph # Related objects

Show commands for: SageMath
sage: E = EllipticCurve("6720.m1")

sage: E.isogeny_class()

## Elliptic curves in class 6720.m

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
6720.m1 6720bm5 [0, -1, 0, -41921, 3317601]  16384
6720.m2 6720bm3 [0, -1, 0, -2721, 48321] [2, 2] 8192
6720.m3 6720bm2 [0, -1, 0, -721, -6479] [2, 2] 4096
6720.m4 6720bm1 [0, -1, 0, -701, -6915]  2048 $$\Gamma_0(N)$$-optimal
6720.m5 6720bm4 [0, -1, 0, 959, -33695]  8192
6720.m6 6720bm6 [0, -1, 0, 4479, 254241]  16384

## Rank

sage: E.rank()

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

## Modular form6720.2.a.m

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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels. 