# Properties

 Label 444675.cs Number of curves $6$ Conductor $444675$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("cs1")

sage: E.isogeny_class()

## Elliptic curves in class 444675.cs

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
444675.cs1 444675cs6 [1, 0, 0, -451967673088, -116952447019390333] [2] 1061683200
444675.cs2 444675cs4 [1, 0, 0, -28247982463, -1827383356887208] [2, 2] 530841600
444675.cs3 444675cs5 [1, 0, 0, -28107909838, -1846402838127583] [2] 1061683200
444675.cs4 444675cs3 [1, 0, 0, -3771588213, 47014048551042] [2] 530841600 $$\Gamma_0(N)$$-optimal*
444675.cs5 444675cs2 [1, 0, 0, -1774256338, -28255403158333] [2, 2] 265420800 $$\Gamma_0(N)$$-optimal*
444675.cs6 444675cs1 [1, 0, 0, 5184787, -1320002849208] [2] 132710400 $$\Gamma_0(N)$$-optimal*
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 0 curves highlighted, and conditionally curve 444675.cs1.

## Rank

sage: E.rank()

The elliptic curves in class 444675.cs have rank $$0$$.

## Complex multiplication

The elliptic curves in class 444675.cs do not have complex multiplication.

## Modular form 444675.2.a.cs

sage: E.q_eigenform(10)

$$q - q^{2} + q^{3} - q^{4} - q^{6} + 3q^{8} + q^{9} - q^{12} + 2q^{13} - q^{16} - 2q^{17} - q^{18} + 4q^{19} + O(q^{20})$$

## 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 LMFDB numbering.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.