# Properties

 Label 55440ct Number of curves $6$ Conductor $55440$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("ct1")

sage: E.isogeny_class()

## Elliptic curves in class 55440ct

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
55440.e6 55440ct1 [0, 0, 0, 5037, -39970222] [2] 491520 $$\Gamma_0(N)$$-optimal
55440.e5 55440ct2 [0, 0, 0, -1723683, -855580318] [2, 2] 983040
55440.e4 55440ct3 [0, 0, 0, -3664083, 1423613522] [2] 1966080
55440.e2 55440ct4 [0, 0, 0, -27442803, -55333820302] [2, 2] 1966080
55440.e3 55440ct5 [0, 0, 0, -27306723, -55909738078] [2] 3932160
55440.e1 55440ct6 [0, 0, 0, -439084803, -3541365261502] [2] 3932160

## Rank

sage: E.rank()

The elliptic curves in class 55440ct have rank $$1$$.

## Complex multiplication

The elliptic curves in class 55440ct do not have complex multiplication.

## Modular form 55440.2.a.ct

sage: E.q_eigenform(10)

$$q - q^{5} - q^{7} - q^{11} - 2q^{13} - 2q^{17} - 4q^{19} + O(q^{20})$$

## 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.

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

## Isogeny graph

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

The vertices are labelled with Cremona labels.