## Fano threefolds with $\rho=5$

ID | $-\mathrm{K}_X^3$ | $\mathrm{h}^{1,2}$ | description | blowups | blowdowns | rational | unirational | moduli | $\mathrm{Aut}^0$ |
---|---|---|---|---|---|---|---|---|---|

5-1 | 28 | 0 |
blowup of 2-29 in the disjoint union of three exceptional lines of the blowup |
4-4, 4-12 | yes | yes | 0 | $\mathbb{G}_{\mathrm{m}}$ | |

5-2 | 36 | 0 |
blowup of 3-25 in the disjoint union of two exceptional lines on the same irreducible component |
4-9, 4-11, 4-12 | yes | yes | 0 | $\mathbb{G}_{\mathrm{m}}\times\mathrm{GL}_2$ | |

5-3 | 36 | 0 |
$\mathbb{P}^1\times\mathrm{Bl}_3\mathbb{P}^2$ |
* | 4-10 | yes | yes | 0 | $\mathrm{PGL}_2\times\mathbb{G}_{\mathrm{m}}^2$ |