# Properties

 Label 40656co Number of curves $6$ Conductor $40656$ CM no Rank $2$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("40656.cd1")

sage: E.isogeny_class()

## Elliptic curves in class 40656co

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
40656.cd5 40656co1 [0, 1, 0, -7784, 581172] [2] 122880 $$\Gamma_0(N)$$-optimal
40656.cd4 40656co2 [0, 1, 0, -162664, 25176116] [2, 2] 245760
40656.cd3 40656co3 [0, 1, 0, -201384, 12243636] [2, 2] 491520
40656.cd1 40656co4 [0, 1, 0, -2602024, 1614663092] [2] 491520
40656.cd6 40656co5 [0, 1, 0, 747256, 95344500] [2] 983040
40656.cd2 40656co6 [0, 1, 0, -1769544, -897916428] [2] 983040

## Rank

sage: E.rank()

The elliptic curves in class 40656co have rank $$2$$.

## Modular form 40656.2.a.cd

sage: E.q_eigenform(10)

$$q + q^{3} - 2q^{5} - q^{7} + q^{9} - 6q^{13} - 2q^{15} - 2q^{17} - 4q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.