# Properties

 Label 416955bs Number of curves $6$ Conductor $416955$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("bs1")

sage: E.isogeny_class()

## Elliptic curves in class 416955bs

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
416955.bs6 416955bs1 [1, 1, 0, 12628, 158660211] [2] 6635520 $$\Gamma_0(N)$$-optimal*
416955.bs5 416955bs2 [1, 1, 0, -4321177, 3394279024] [2, 2] 13271040 $$\Gamma_0(N)$$-optimal*
416955.bs2 416955bs3 [1, 1, 0, -68797582, 219609455551] [2, 2] 26542080 $$\Gamma_0(N)$$-optimal*
416955.bs4 416955bs4 [1, 1, 0, -9185652, -5654617371] [2] 26542080
416955.bs1 416955bs5 [1, 1, 0, -1100761207, 14056384132276] [2] 53084160 $$\Gamma_0(N)$$-optimal*
416955.bs3 416955bs6 [1, 1, 0, -68456437, 221895604654] [2] 53084160
*optimality has not been determined rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 4 curves highlighted, and conditionally curve 416955bs1.

## Rank

sage: E.rank()

The elliptic curves in class 416955bs have rank $$0$$.

## Complex multiplication

The elliptic curves in class 416955bs do not have complex multiplication.

## Modular form 416955.2.a.bs

sage: E.q_eigenform(10)

$$q + q^{2} - q^{3} - q^{4} + q^{5} - q^{6} + q^{7} - 3q^{8} + q^{9} + q^{10} - q^{11} + q^{12} + 2q^{13} + q^{14} - q^{15} - q^{16} + 2q^{17} + q^{18} + O(q^{20})$$

## 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.