# Properties

 Label 6720bw Number of curves 8 Conductor 6720 CM no Rank 1 Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("6720.bi1")

sage: E.isogeny_class()

## Elliptic curves in class 6720bw

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
6720.bi7 6720bw1 [0, 1, 0, -31841, -2196705] [2] 18432 $$\Gamma_0(N)$$-optimal
6720.bi6 6720bw2 [0, 1, 0, -36961, -1448161] [2, 2] 36864
6720.bi5 6720bw3 [0, 1, 0, -94241, 8431455] [2] 55296
6720.bi4 6720bw4 [0, 1, 0, -278881, 55596575] [2] 73728
6720.bi8 6720bw5 [0, 1, 0, 123039, -10504161] [2] 73728
6720.bi2 6720bw6 [0, 1, 0, -1404961, 640460639] [2, 2] 110592
6720.bi1 6720bw7 [0, 1, 0, -22478881, 41013876575] [2] 221184
6720.bi3 6720bw8 [0, 1, 0, -1302561, 737883999] [2] 221184

## Rank

sage: E.rank()

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

## Modular form6720.2.a.bi

sage: E.q_eigenform(10)

$$q + q^{3} - q^{5} - q^{7} + q^{9} - 2q^{13} - q^{15} - 6q^{17} + 8q^{19} + O(q^{20})$$

## 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.

$$\left(\begin{array}{rrrrrrrr} 1 & 2 & 3 & 4 & 4 & 6 & 12 & 12 \\ 2 & 1 & 6 & 2 & 2 & 3 & 6 & 6 \\ 3 & 6 & 1 & 12 & 12 & 2 & 4 & 4 \\ 4 & 2 & 12 & 1 & 4 & 6 & 3 & 12 \\ 4 & 2 & 12 & 4 & 1 & 6 & 12 & 3 \\ 6 & 3 & 2 & 6 & 6 & 1 & 2 & 2 \\ 12 & 6 & 4 & 3 & 12 & 2 & 1 & 4 \\ 12 & 6 & 4 & 12 & 3 & 2 & 4 & 1 \end{array}\right)$$

## Isogeny graph

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

The vertices are labelled with Cremona labels.