# Properties

 Label 6864r Number of curves $6$ Conductor $6864$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("6864.e1")

sage: E.isogeny_class()

## Elliptic curves in class 6864r

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
6864.e5 6864r1 [0, -1, 0, -384, -4032] [2] 4096 $$\Gamma_0(N)$$-optimal
6864.e4 6864r2 [0, -1, 0, -6864, -216576] [2, 2] 8192
6864.e1 6864r3 [0, -1, 0, -109824, -13972032] [2] 16384
6864.e3 6864r4 [0, -1, 0, -7584, -167616] [2, 4] 16384
6864.e2 6864r5 [0, -1, 0, -48144, 3953280] [4] 32768
6864.e6 6864r6 [0, -1, 0, 21456, -1166592] [4] 32768

## Rank

sage: E.rank()

The elliptic curves in class 6864r have rank $$0$$.

## Modular form6864.2.a.e

sage: E.q_eigenform(10)

$$q - q^{3} - 2q^{5} + q^{9} + q^{11} + q^{13} + 2q^{15} - 6q^{17} + 4q^{19} + O(q^{20})$$

## 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.