s2=(-4 α+2 (α^2)+(α^3)-4 β+4 α β-(α^2) β+2 (β^2)-α (β^2)+(β^3))/(2 (-4+4 α+(α^2)+4 β-2 α β+(β^2)));

s3=(128-64 α-224 (α^2)+144 (α^3)+56 (α^4)-76 (α^5)+30 (α^6)+3 (α^7)-64 β-64 α β+112 (α^2) β-96 (α^3) β+36 (α^4) β+76 (α^5) β-15 (α^6) β-224 (β^2)+112 α (β^2)+80 (α^2) (β^2)+40 (α^3) (β^2)-62 (α^4) (β^2)+27 (α^5) (β^2)+144 (β^3)-96 α (β^3)+40 (α^2) (β^3)-88 (α^3) (β^3)-15 (α^4) (β^3)+56 (β^4)+36 α (β^4)-62 (α^2) (β^4)-15 (α^3) (β^4)-76 (β^5)+76 α (β^5)+27 (α^2) (β^5)+30 (β^6)-15 α (β^6)+3 (β^7))/(6 (-64-64 α+112 (α^2)+32 (α^3)-28 (α^4)+12 (α^5)+(α^6)-64 β+32 α β-32 (α^2) β+48 (α^3) β+28 (α^4) β-6 (α^5) β+112 (β^2)-32 α (β^2)-40 (α^2) (β^2)-40 (α^3) (β^2)+15 (α^4) (β^2)+32 (β^3)+48 α (β^3)-40 (α^2) (β^3)-20 (α^3) (β^3)-28 (β^4)+28 α (β^4)+15 (α^2) (β^4)+12 (β^5)-6 α (β^5)+(β^6)));

s4=(4096 α-2048 (α^2)-8192 (α^3)+3584 (α^4)+5888 (α^5)-256 (α^6)-3072 (α^7)-1600 (α^8)+1392 (α^9)+280 (α^10)-96 (α^11)+22 (α^12)+(α^13)+4096 β-4096 α β+4096 (α^3) β-9472 (α^4) β+4608 (α^5) β+4096 (α^6) β-4352 (α^7) β+1008 (α^8) β-400 (α^9) β+320 (α^10) β+96 (α^11) β-11 (α^12) β-2048 (β^2)+1024 (α^2) (β^2)+3584 (α^3) (β^2)+256 (α^4) (β^2)-10240 (α^5) (β^2)+2816 (α^6) (β^2)+6848 (α^7) (β^2)-1480 (α^8) (β^2)-576 (α^9) (β^2)-292 (α^10) (β^2)+54 (α^11) (β^2)-8192 (β^3)+4096 α (β^3)+3584 (α^2) (β^3)-9216 (α^3) (β^3)+9216 (α^4) (β^3)+4352 (α^5) (β^3)-7744 (α^6) (β^3)+2880 (α^7) (β^3)+1440 (α^8) (β^3)-416 (α^9) (β^3)-154 (α^10) (β^3)+3584 (β^4)-9472 α (β^4)+256 (α^2) (β^4)+9216 (α^3) (β^4)-2432 (α^4) (β^4)-1504 (α^5) (β^4)+1200 (α^6) (β^4)-2624 (α^7) (β^4)+1226 (α^8) (β^4)+275 (α^9) (β^4)+5888 (β^5)+4608 α (β^5)-10240 (α^2) (β^5)+4352 (α^3) (β^5)-1504 (α^4) (β^5)-4960 (α^5) (β^5)+1536 (α^6) (β^5)+320 (α^7) (β^5)-297 (α^8) (β^5)-256 (β^6)+4096 α (β^6)+2816 (α^2) (β^6)-7744 (α^3) (β^6)+1200 (α^4) (β^6)+1536 (α^5) (β^6)-1912 (α^6) (β^6)+132 (α^7) (β^6)-3072 (β^7)-4352 α (β^7)+6848 (α^2) (β^7)+2880 (α^3) (β^7)-2624 (α^4) (β^7)+320 (α^5) (β^7)+132 (α^6) (β^7)-1600 (β^8)+1008 α (β^8)-1480 (α^2) (β^8)+1440 (α^3) (β^8)+1226 (α^4) (β^8)-297 (α^5) (β^8)+1392 (β^9)-400 α (β^9)-576 (α^2) (β^9)-416 (α^3) (β^9)+275 (α^4) (β^9)+280 (β^10)+320 α (β^10)-292 (α^2) (β^10)-154 (α^3) (β^10)-96 (β^11)+96 α (β^11)+54 (α^2) (β^11)+22 (β^12)-11 α (β^12)+(β^13))/(2 (4096-8192 α-10240 (α^2)+14336 (α^3)+7936 (α^4)-5120 (α^5)-3840 (α^6)-1280 (α^7)+2032 (α^8)+352 (α^9)-104 (α^10)+24 (α^11)+(α^12)-8192 β+4096 α β+2048 (α^2) β-15360 (α^3) β+15360 (α^4) β+8704 (α^5) β-7936 (α^6) β+1152 (α^7) β-416 (α^8) β+464 (α^9) β+104 (α^10) β-12 (α^11) β-10240 (β^2)+2048 α (β^2)+14848 (α^2) (β^2)-10240 (α^3) (β^2)-16640 (α^4) (β^2)+15104 (α^5) (β^2)+9792 (α^6) (β^2)-3200 (α^7) (β^2)-1096 (α^8) (β^2)-376 (α^9) (β^2)+66 (α^10) (β^2)+14336 (β^3)-15360 α (β^3)-10240 (α^2) (β^3)+23552 (α^3) (β^3)-5888 (α^4) (β^3)-16512 (α^5) (β^3)+8576 (α^6) (β^3)+2496 (α^7) (β^3)-520 (α^8) (β^3)-220 (α^9) (β^3)+7936 (β^4)+15360 α (β^4)-16640 (α^2) (β^4)-5888 (α^3) (β^4)+7072 (α^4) (β^4)-5312 (α^5) (β^4)-4944 (α^6) (β^4)+2416 (α^7) (β^4)+495 (α^8) (β^4)-5120 (β^5)+8704 α (β^5)+15104 (α^2) (β^5)-16512 (α^3) (β^5)-5312 (α^4) (β^5)+6368 (α^5) (β^5)-1648 (α^6) (β^5)-792 (α^7) (β^5)-3840 (β^6)-7936 α (β^6)+9792 (α^2) (β^6)+8576 (α^3) (β^6)-4944 (α^4) (β^6)-1648 (α^5) (β^6)+924 (α^6) (β^6)-1280 (β^7)+1152 α (β^7)-3200 (α^2) (β^7)+2496 (α^3) (β^7)+2416 (α^4) (β^7)-792 (α^5) (β^7)+2032 (β^8)-416 α (β^8)-1096 (α^2) (β^8)-520 (α^3) (β^8)+495 (α^4) (β^8)+352 (β^9)+464 α (β^9)-376 (α^2) (β^9)-220 (α^3) (β^9)-104 (β^10)+104 α (β^10)+66 (α^2) (β^10)+24 (β^11)-12 α (β^11)+(β^12)));

Together[1/2Det[{{σ0,σ2},{σ1,σ3}}]/Det[{{σ0,σ1},{σ1,σ2}}]]

Together[1/3Det[{{σ0,σ1,σ3},{σ1,σ2,σ4},{σ2,σ3,σ5}}]/Det[{{σ0,σ1,σ2},{σ1,σ2,σ3},{σ2,σ3,σ4}}]]

Together[1/4Det[{{σ0,σ1,σ2,σ4},{σ1,σ2,σ3,σ5},{σ2,σ3,σ4,σ6},{σ3,σ4,σ5,σ7}}]/Det[{{σ0,σ1,σ2,σ3},{σ1,σ2,σ3,σ4},{σ2,σ3,σ4,σ5},{σ3,σ4,σ5,σ6}}]]