# Properties

 Label 417450.fo Number of curves $6$ Conductor $417450$ CM no Rank $0$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("417450.fo1")

sage: E.isogeny_class()

## Elliptic curves in class 417450.fo

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
417450.fo1 417450fo4 [1, 1, 1, -333960063, -2349176461719] [2] 47185920
417450.fo2 417450fo5 [1, 1, 1, -78135813, 227684489781] [2] 94371840 $$\Gamma_0(N)$$-optimal*
417450.fo3 417450fo3 [1, 1, 1, -21417063, -34696447719] [2, 2] 47185920 $$\Gamma_0(N)$$-optimal*
417450.fo4 417450fo2 [1, 1, 1, -20872563, -36712186719] [2, 2] 23592960 $$\Gamma_0(N)$$-optimal*
417450.fo5 417450fo1 [1, 1, 1, -1270563, -605302719] [2] 11796480 $$\Gamma_0(N)$$-optimal*
417450.fo6 417450fo6 [1, 1, 1, 26589687, -168059199219] [2] 94371840
*optimality has not been proved rigorously for conductors over 400000. In this case the optimal curve is certainly one of the 0 curves highlighted, and conditionally curve 417450.fo5.

## Rank

sage: E.rank()

The elliptic curves in class 417450.fo have rank $$0$$.

## Modular form 417450.2.a.fo

sage: E.q_eigenform(10)

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

## 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.

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

## Isogeny graph

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

The vertices are labelled with LMFDB labels.