# Properties

 Label 418275.cg Number of curves $6$ Conductor $418275$ CM no Rank $0$ Graph

# Learn more about

Show commands for: SageMath
sage: E = EllipticCurve("418275.cg1")

sage: E.isogeny_class()

## Elliptic curves in class 418275.cg

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
418275.cg1 418275cg4 [1, -1, 0, -261004392, -1622939671859] [2] 44040192
418275.cg2 418275cg5 [1, -1, 0, -114418017, 456186144766] [2] 88080384 $$\Gamma_0(N)$$-optimal*
418275.cg3 418275cg3 [1, -1, 0, -18024642, -19707947609] [2, 2] 44040192 $$\Gamma_0(N)$$-optimal*
418275.cg4 418275cg2 [1, -1, 0, -16313517, -25352948984] [2, 2] 22020096 $$\Gamma_0(N)$$-optimal*
418275.cg5 418275cg1 [1, -1, 0, -913392, -481747109] [2] 11010048 $$\Gamma_0(N)$$-optimal*
418275.cg6 418275cg6 [1, -1, 0, 50990733, -134342485484] [2] 88080384
*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 418275.cg5.

## Rank

sage: E.rank()

The elliptic curves in class 418275.cg have rank $$0$$.

## Modular form 418275.2.a.cg

sage: E.q_eigenform(10)

$$q + q^{2} - q^{4} - 3q^{8} - q^{11} - q^{16} - 6q^{17} + 4q^{19} + O(q^{20})$$

## 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 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.