# Properties

 Label 6720f Number of curves 8 Conductor 6720 CM no Rank 0 Graph

# Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("6720.n1")

sage: E.isogeny_class()

## Elliptic curves in class 6720f

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
6720.n7 6720f1 [0, -1, 0, -31841, 2196705] [2] 18432 $$\Gamma_0(N)$$-optimal
6720.n6 6720f2 [0, -1, 0, -36961, 1448161] [2, 2] 36864
6720.n5 6720f3 [0, -1, 0, -94241, -8431455] [2] 55296
6720.n4 6720f4 [0, -1, 0, -278881, -55596575] [2] 73728
6720.n8 6720f5 [0, -1, 0, 123039, 10504161] [2] 73728
6720.n2 6720f6 [0, -1, 0, -1404961, -640460639] [2, 2] 110592
6720.n1 6720f7 [0, -1, 0, -22478881, -41013876575] [2] 221184
6720.n3 6720f8 [0, -1, 0, -1302561, -737883999] [2] 221184

## Rank

sage: E.rank()

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

## Modular form6720.2.a.n

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.