# Properties

 Label 163170.ca Number of curves $6$ Conductor $163170$ CM no Rank $1$ Graph

# Related objects

Show commands for: SageMath
sage: E = EllipticCurve("163170.ca1")

sage: E.isogeny_class()

## Elliptic curves in class 163170.ca

sage: E.isogeny_class().curves

LMFDB label Cremona label Weierstrass coefficients Torsion structure Modular degree Optimality
163170.ca1 163170de3 [1, -1, 0, -150383214, 709855836948] [2] 21233664
163170.ca2 163170de6 [1, -1, 0, -133678134, -592273012500] [2] 42467328
163170.ca3 163170de4 [1, -1, 0, -12932334, 2013665940] [2, 2] 21233664
163170.ca4 163170de2 [1, -1, 0, -9404334, 11079920340] [2, 2] 10616832
163170.ca5 163170de1 [1, -1, 0, -372654, 301513428] [2] 5308416 $$\Gamma_0(N)$$-optimal
163170.ca6 163170de5 [1, -1, 0, 51365466, 16017726780] [2] 42467328

## Rank

sage: E.rank()

The elliptic curves in class 163170.ca have rank $$1$$.

## Modular form 163170.2.a.ca

sage: E.q_eigenform(10)

$$q - q^{2} + q^{4} + q^{5} - q^{8} - q^{10} - 4q^{11} + 2q^{13} + q^{16} + 2q^{17} - 4q^{19} + O(q^{20})$$

## 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 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.