LMFDB - Number field 32.0.24000959919026880122072334336000000000000000000000000.2

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 2*x^31 + x^30 - 10*x^29 - 20*x^28 - 136*x^27 + 489*x^26 + 412*x^25 + 1455*x^24 - 380*x^23 + 20765*x^22 - 113020*x^21 + 22444*x^20 - 71630*x^19 + 273805*x^18 - 1917530*x^17 + 8820755*x^16 - 6803856*x^15 - 3050102*x^14 + 14422848*x^13 - 16249472*x^12 + 4718368*x^11 + 18224552*x^10 - 33044576*x^9 + 20240880*x^8 + 9340480*x^7 + 2473504*x^6 + 461696*x^5 - 5632*x^4 + 15872*x^3 + 6272*x^2 + 1536*x + 256)

gp: K = bnfinit(y^32 - 2*y^31 + y^30 - 10*y^29 - 20*y^28 - 136*y^27 + 489*y^26 + 412*y^25 + 1455*y^24 - 380*y^23 + 20765*y^22 - 113020*y^21 + 22444*y^20 - 71630*y^19 + 273805*y^18 - 1917530*y^17 + 8820755*y^16 - 6803856*y^15 - 3050102*y^14 + 14422848*y^13 - 16249472*y^12 + 4718368*y^11 + 18224552*y^10 - 33044576*y^9 + 20240880*y^8 + 9340480*y^7 + 2473504*y^6 + 461696*y^5 - 5632*y^4 + 15872*y^3 + 6272*y^2 + 1536*y + 256, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 - 2*x^31 + x^30 - 10*x^29 - 20*x^28 - 136*x^27 + 489*x^26 + 412*x^25 + 1455*x^24 - 380*x^23 + 20765*x^22 - 113020*x^21 + 22444*x^20 - 71630*x^19 + 273805*x^18 - 1917530*x^17 + 8820755*x^16 - 6803856*x^15 - 3050102*x^14 + 14422848*x^13 - 16249472*x^12 + 4718368*x^11 + 18224552*x^10 - 33044576*x^9 + 20240880*x^8 + 9340480*x^7 + 2473504*x^6 + 461696*x^5 - 5632*x^4 + 15872*x^3 + 6272*x^2 + 1536*x + 256);

oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^32 - 2*x^31 + x^30 - 10*x^29 - 20*x^28 - 136*x^27 + 489*x^26 + 412*x^25 + 1455*x^24 - 380*x^23 + 20765*x^22 - 113020*x^21 + 22444*x^20 - 71630*x^19 + 273805*x^18 - 1917530*x^17 + 8820755*x^16 - 6803856*x^15 - 3050102*x^14 + 14422848*x^13 - 16249472*x^12 + 4718368*x^11 + 18224552*x^10 - 33044576*x^9 + 20240880*x^8 + 9340480*x^7 + 2473504*x^6 + 461696*x^5 - 5632*x^4 + 15872*x^3 + 6272*x^2 + 1536*x + 256)

$ x^{32} - 2 x^{31} + x^{30} - 10 x^{29} - 20 x^{28} - 136 x^{27} + 489 x^{26} + 412 x^{25} + 1455 x^{24} + \cdots + 256 $ x^32 - 2*x^31 + x^30 - 10*x^29 - 20*x^28 - 136*x^27 + 489*x^26 + 412*x^25 + 1455*x^24 - 380*x^23 + 20765*x^22 - 113020*x^21 + 22444*x^20 - 71630*x^19 + 273805*x^18 - 1917530*x^17 + 8820755*x^16 - 6803856*x^15 - 3050102*x^14 + 14422848*x^13 - 16249472*x^12 + 4718368*x^11 + 18224552*x^10 - 33044576*x^9 + 20240880*x^8 + 9340480*x^7 + 2473504*x^6 + 461696*x^5 - 5632*x^4 + 15872*x^3 + 6272*x^2 + 1536*x + 256

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

oscar: defining_polynomial(K)

Invariants

Degree:	$32$	sage: K.degree() gp: poldegree(K.pol) magma: Degree(K); oscar: degree(K)
Signature:	$[0, 16]$	sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K)
Discriminant:	$24000959919026880122072334336000000000000000000000000$ 24000959919026880122072334336000000000000000000000000 $\medspace = 2^{48}\cdot 3^{16}\cdot 5^{24}\cdot 7^{16}$	sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK)
Root discriminant:	$43.34$	sage: (K.disc().abs())^(1./K.degree()) gp: abs(K.disc)^(1/poldegree(K.pol)) magma: Abs(Discriminant(OK))^(1/Degree(K)); oscar: (1.0 * dK)^(1/degree(K))
Galois root discriminant:	$2^{3/2}3^{1/2}5^{3/4}7^{1/2}\approx 43.339325111263825$
Ramified primes:	$2$, $3$, $5$, $7$ 2, 3, 5, 7	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK)))
Discriminant root field:	$\Q$
$\card{ \Gal(K/\Q) }$:	$32$	sage: K.automorphisms() magma: Automorphisms(K); oscar: automorphisms(K)
This field is Galois and abelian over $\Q$.
Conductor:	$840=2^{3}\cdot 3\cdot 5\cdot 7$
Dirichlet character group:	$\lbrace$$\chi_{840}(1,·)$, $\chi_{840}(517,·)$, $\chi_{840}(769,·)$, $\chi_{840}(13,·)$, $\chi_{840}(533,·)$, $\chi_{840}(281,·)$, $\chi_{840}(797,·)$, $\chi_{840}(673,·)$, $\chi_{840}(421,·)$, $\chi_{840}(41,·)$, $\chi_{840}(29,·)$, $\chi_{840}(433,·)$, $\chi_{840}(181,·)$, $\chi_{840}(701,·)$, $\chi_{840}(629,·)$, $\chi_{840}(449,·)$, $\chi_{840}(197,·)$, $\chi_{840}(713,·)$, $\chi_{840}(589,·)$, $\chi_{840}(461,·)$, $\chi_{840}(209,·)$, $\chi_{840}(601,·)$, $\chi_{840}(349,·)$, $\chi_{840}(293,·)$, $\chi_{840}(97,·)$, $\chi_{840}(337,·)$, $\chi_{840}(617,·)$, $\chi_{840}(113,·)$, $\chi_{840}(757,·)$, $\chi_{840}(169,·)$, $\chi_{840}(377,·)$, $\chi_{840}(253,·)$$\rbrace$
This is a CM field.
Reflex fields:	unavailable$^{32768}$

Integral basis (with respect to field generator $a$)

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{2}a^{18}-\frac{1}{2}a^{16}-\frac{1}{2}a^{12}-\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{19}-\frac{1}{2}a^{17}-\frac{1}{2}a^{13}-\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{20}-\frac{1}{4}a^{18}-\frac{1}{2}a^{16}+\frac{1}{4}a^{14}-\frac{1}{2}a^{13}+\frac{1}{4}a^{12}-\frac{1}{2}a^{11}-\frac{1}{4}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}+\frac{1}{4}a^{6}+\frac{1}{4}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{21}-\frac{1}{4}a^{19}-\frac{1}{2}a^{17}+\frac{1}{4}a^{15}-\frac{1}{2}a^{14}+\frac{1}{4}a^{13}-\frac{1}{2}a^{12}-\frac{1}{4}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}+\frac{1}{4}a^{7}+\frac{1}{4}a^{5}-\frac{1}{2}a^{4}$, $\frac{1}{8}a^{22}-\frac{1}{8}a^{20}-\frac{1}{4}a^{18}-\frac{3}{8}a^{16}-\frac{1}{4}a^{15}+\frac{1}{8}a^{14}-\frac{1}{4}a^{13}-\frac{1}{8}a^{12}+\frac{1}{4}a^{11}+\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{3}{8}a^{8}-\frac{1}{2}a^{7}+\frac{1}{8}a^{6}-\frac{1}{4}a^{5}-\frac{1}{2}a^{4}$, $\frac{1}{8}a^{23}-\frac{1}{8}a^{21}-\frac{1}{4}a^{19}-\frac{3}{8}a^{17}-\frac{1}{4}a^{16}+\frac{1}{8}a^{15}-\frac{1}{4}a^{14}-\frac{1}{8}a^{13}+\frac{1}{4}a^{12}+\frac{1}{4}a^{11}-\frac{1}{4}a^{10}-\frac{3}{8}a^{9}-\frac{1}{2}a^{8}+\frac{1}{8}a^{7}-\frac{1}{4}a^{6}-\frac{1}{2}a^{5}$, $\frac{1}{336}a^{24}+\frac{1}{56}a^{23}+\frac{1}{336}a^{22}-\frac{17}{168}a^{21}+\frac{1}{21}a^{19}-\frac{19}{336}a^{18}-\frac{17}{84}a^{17}-\frac{113}{336}a^{16}+\frac{11}{84}a^{15}+\frac{1}{336}a^{14}+\frac{5}{28}a^{13}+\frac{5}{42}a^{12}-\frac{1}{24}a^{11}+\frac{27}{112}a^{10}-\frac{1}{168}a^{9}-\frac{15}{112}a^{8}-\frac{1}{21}a^{7}+\frac{3}{8}a^{6}-\frac{3}{14}a^{5}-\frac{13}{84}a^{4}+\frac{3}{7}a^{3}-\frac{4}{21}a+\frac{1}{21}$, $\frac{1}{336}a^{25}+\frac{1}{48}a^{23}+\frac{1}{168}a^{22}-\frac{1}{56}a^{21}-\frac{13}{168}a^{20}-\frac{31}{336}a^{19}-\frac{19}{168}a^{18}-\frac{167}{336}a^{17}-\frac{10}{21}a^{16}-\frac{137}{336}a^{15}+\frac{1}{28}a^{14}+\frac{29}{168}a^{13}+\frac{31}{84}a^{12}+\frac{55}{112}a^{11}-\frac{19}{42}a^{10}+\frac{31}{112}a^{9}-\frac{5}{42}a^{8}-\frac{3}{14}a^{7}+\frac{23}{56}a^{6}-\frac{5}{42}a^{5}-\frac{1}{7}a^{4}+\frac{3}{7}a^{3}-\frac{4}{21}a^{2}+\frac{4}{21}a-\frac{2}{7}$, $\frac{1}{8555232}a^{26}-\frac{379}{1069404}a^{25}+\frac{9139}{8555232}a^{24}-\frac{19281}{356468}a^{23}+\frac{202837}{4277616}a^{22}-\frac{15367}{1069404}a^{21}-\frac{244801}{2851744}a^{20}-\frac{166055}{1425872}a^{19}+\frac{270317}{8555232}a^{18}-\frac{659415}{1425872}a^{17}-\frac{425519}{2851744}a^{16}-\frac{1110559}{4277616}a^{15}-\frac{808657}{4277616}a^{14}-\frac{933689}{4277616}a^{13}-\frac{981059}{8555232}a^{12}-\frac{395105}{2138808}a^{11}-\frac{3022619}{8555232}a^{10}-\frac{421313}{4277616}a^{9}+\frac{230263}{1069404}a^{8}-\frac{13919}{89117}a^{7}+\frac{448585}{1069404}a^{6}-\frac{10763}{267351}a^{5}-\frac{1343}{89117}a^{4}+\frac{5230}{267351}a^{3}+\frac{7930}{89117}a^{2}+\frac{55931}{267351}a+\frac{627}{89117}$, $\frac{1}{8555232}a^{27}+\frac{7897}{8555232}a^{25}+\frac{197}{712936}a^{24}-\frac{26207}{712936}a^{23}-\frac{4277}{152772}a^{22}-\frac{22819}{295008}a^{21}-\frac{323467}{4277616}a^{20}-\frac{681733}{8555232}a^{19}+\frac{73859}{4277616}a^{18}+\frac{2402585}{8555232}a^{17}-\frac{2048293}{4277616}a^{16}-\frac{37648}{89117}a^{15}-\frac{1885979}{4277616}a^{14}+\frac{213835}{2851744}a^{13}-\frac{598351}{2138808}a^{12}+\frac{1293571}{8555232}a^{11}-\frac{341039}{4277616}a^{10}-\frac{535489}{4277616}a^{9}-\frac{165115}{712936}a^{8}+\frac{54245}{267351}a^{7}-\frac{308983}{712936}a^{6}+\frac{24565}{50924}a^{5}+\frac{187627}{534702}a^{4}+\frac{23917}{534702}a^{3}+\frac{22032}{89117}a^{2}-\frac{30440}{267351}a+\frac{63359}{267351}$, $\frac{1}{17880434880}a^{28}+\frac{211}{8940217440}a^{27}-\frac{11}{232213440}a^{26}+\frac{5553337}{8940217440}a^{25}-\frac{614671}{447010872}a^{24}-\frac{6246727}{127717392}a^{23}-\frac{187532801}{5960144960}a^{22}-\frac{4820043}{78422960}a^{21}+\frac{673921309}{5960144960}a^{20}+\frac{12089465}{223505436}a^{19}-\frac{863782915}{3576086976}a^{18}+\frac{414498767}{1117527180}a^{17}-\frac{22992538}{55876359}a^{16}-\frac{55063451}{812747040}a^{15}+\frac{285526855}{1192028992}a^{14}+\frac{260431565}{596014496}a^{13}+\frac{2250470049}{5960144960}a^{12}+\frac{736765027}{1490036240}a^{11}-\frac{902218109}{8940217440}a^{10}-\frac{426886445}{894021744}a^{9}-\frac{404586643}{894021744}a^{8}-\frac{9824039}{50796690}a^{7}+\frac{1022203}{319293480}a^{6}-\frac{40143434}{279381795}a^{5}-\frac{9155921}{38535420}a^{4}-\frac{23811454}{55876359}a^{3}+\frac{81393091}{186254530}a^{2}+\frac{3394115}{55876359}a+\frac{1992179}{13303895}$, $\frac{1}{51\!\cdots\!00}a^{29}+\frac{21\!\cdots\!01}{12\!\cdots\!00}a^{28}+\frac{17\!\cdots\!91}{46\!\cdots\!00}a^{27}+\frac{38\!\cdots\!11}{92\!\cdots\!00}a^{26}-\frac{30\!\cdots\!49}{32\!\cdots\!00}a^{25}-\frac{19\!\cdots\!81}{17\!\cdots\!20}a^{24}+\frac{86\!\cdots\!37}{51\!\cdots\!00}a^{23}-\frac{32\!\cdots\!69}{12\!\cdots\!00}a^{22}+\frac{17\!\cdots\!91}{51\!\cdots\!00}a^{21}+\frac{19\!\cdots\!59}{25\!\cdots\!00}a^{20}-\frac{36\!\cdots\!87}{20\!\cdots\!84}a^{19}-\frac{40\!\cdots\!59}{25\!\cdots\!00}a^{18}+\frac{32\!\cdots\!37}{12\!\cdots\!00}a^{17}-\frac{49\!\cdots\!57}{40\!\cdots\!00}a^{16}+\frac{79\!\cdots\!79}{17\!\cdots\!00}a^{15}+\frac{41\!\cdots\!23}{24\!\cdots\!60}a^{14}+\frac{12\!\cdots\!27}{51\!\cdots\!00}a^{13}-\frac{38\!\cdots\!33}{86\!\cdots\!00}a^{12}+\frac{98\!\cdots\!57}{36\!\cdots\!00}a^{11}+\frac{59\!\cdots\!07}{45\!\cdots\!00}a^{10}-\frac{80\!\cdots\!17}{25\!\cdots\!80}a^{9}-\frac{24\!\cdots\!77}{11\!\cdots\!00}a^{8}-\frac{14\!\cdots\!69}{32\!\cdots\!00}a^{7}-\frac{70\!\cdots\!27}{46\!\cdots\!00}a^{6}-\frac{30\!\cdots\!94}{26\!\cdots\!25}a^{5}+\frac{73\!\cdots\!59}{53\!\cdots\!50}a^{4}-\frac{12\!\cdots\!26}{80\!\cdots\!75}a^{3}-\frac{62\!\cdots\!03}{16\!\cdots\!50}a^{2}-\frac{18\!\cdots\!16}{80\!\cdots\!75}a-\frac{30\!\cdots\!24}{73\!\cdots\!25}$, $\frac{1}{10\!\cdots\!00}a^{30}-\frac{15\!\cdots\!33}{68\!\cdots\!80}a^{28}-\frac{67\!\cdots\!23}{92\!\cdots\!60}a^{27}+\frac{12\!\cdots\!37}{86\!\cdots\!60}a^{26}-\frac{14\!\cdots\!27}{12\!\cdots\!00}a^{25}-\frac{69\!\cdots\!83}{10\!\cdots\!00}a^{24}-\frac{29\!\cdots\!49}{51\!\cdots\!00}a^{23}-\frac{24\!\cdots\!73}{10\!\cdots\!00}a^{22}+\frac{91\!\cdots\!27}{17\!\cdots\!00}a^{21}+\frac{12\!\cdots\!97}{10\!\cdots\!00}a^{20}-\frac{11\!\cdots\!09}{51\!\cdots\!00}a^{19}-\frac{14\!\cdots\!11}{25\!\cdots\!00}a^{18}-\frac{15\!\cdots\!59}{10\!\cdots\!20}a^{17}-\frac{63\!\cdots\!23}{13\!\cdots\!00}a^{16}+\frac{17\!\cdots\!83}{64\!\cdots\!00}a^{15}-\frac{13\!\cdots\!51}{34\!\cdots\!00}a^{14}+\frac{14\!\cdots\!69}{93\!\cdots\!20}a^{13}-\frac{10\!\cdots\!67}{51\!\cdots\!00}a^{12}-\frac{16\!\cdots\!49}{51\!\cdots\!60}a^{11}+\frac{24\!\cdots\!47}{64\!\cdots\!00}a^{10}-\frac{55\!\cdots\!57}{16\!\cdots\!50}a^{9}+\frac{19\!\cdots\!03}{12\!\cdots\!00}a^{8}-\frac{18\!\cdots\!73}{39\!\cdots\!80}a^{7}-\frac{74\!\cdots\!83}{21\!\cdots\!00}a^{6}+\frac{64\!\cdots\!54}{80\!\cdots\!75}a^{5}-\frac{26\!\cdots\!49}{53\!\cdots\!50}a^{4}+\frac{14\!\cdots\!71}{80\!\cdots\!75}a^{3}-\frac{19\!\cdots\!47}{16\!\cdots\!50}a^{2}-\frac{30\!\cdots\!09}{80\!\cdots\!75}a-\frac{16\!\cdots\!04}{11\!\cdots\!25}$, $\frac{1}{10\!\cdots\!00}a^{31}-\frac{1}{10\!\cdots\!00}a^{29}+\frac{75\!\cdots\!91}{73\!\cdots\!00}a^{28}-\frac{96\!\cdots\!63}{51\!\cdots\!00}a^{27}+\frac{57\!\cdots\!33}{17\!\cdots\!00}a^{26}-\frac{83\!\cdots\!57}{14\!\cdots\!00}a^{25}-\frac{57\!\cdots\!69}{46\!\cdots\!00}a^{24}-\frac{67\!\cdots\!79}{20\!\cdots\!40}a^{23}-\frac{53\!\cdots\!49}{12\!\cdots\!00}a^{22}+\frac{96\!\cdots\!71}{10\!\cdots\!00}a^{21}+\frac{45\!\cdots\!13}{36\!\cdots\!00}a^{20}+\frac{90\!\cdots\!53}{51\!\cdots\!00}a^{19}-\frac{34\!\cdots\!69}{36\!\cdots\!00}a^{18}+\frac{16\!\cdots\!27}{34\!\cdots\!00}a^{17}+\frac{43\!\cdots\!09}{12\!\cdots\!40}a^{16}+\frac{71\!\cdots\!97}{20\!\cdots\!40}a^{15}+\frac{48\!\cdots\!57}{12\!\cdots\!40}a^{14}+\frac{10\!\cdots\!23}{21\!\cdots\!00}a^{13}+\frac{76\!\cdots\!59}{46\!\cdots\!00}a^{12}+\frac{10\!\cdots\!53}{21\!\cdots\!00}a^{11}-\frac{11\!\cdots\!37}{12\!\cdots\!00}a^{10}-\frac{14\!\cdots\!89}{43\!\cdots\!00}a^{9}+\frac{14\!\cdots\!17}{43\!\cdots\!00}a^{8}-\frac{50\!\cdots\!49}{12\!\cdots\!40}a^{7}-\frac{70\!\cdots\!37}{32\!\cdots\!00}a^{6}-\frac{45\!\cdots\!01}{16\!\cdots\!55}a^{5}-\frac{41\!\cdots\!87}{23\!\cdots\!50}a^{4}+\frac{24\!\cdots\!53}{53\!\cdots\!50}a^{3}-\frac{91\!\cdots\!13}{27\!\cdots\!75}a^{2}+\frac{19\!\cdots\!12}{23\!\cdots\!65}a-\frac{18\!\cdots\!43}{80\!\cdots\!75}$ 1, a, a^2, a^3, a^4, a^5, a^6, a^7, a^8, a^9, a^10, a^11, a^12, a^13, a^14, a^15, a^16, a^17, 1/2*a^18 - 1/2*a^16 - 1/2*a^12 - 1/2*a^10 - 1/2*a^8 - 1/2*a^4 - 1/2*a^2, 1/2*a^19 - 1/2*a^17 - 1/2*a^13 - 1/2*a^11 - 1/2*a^9 - 1/2*a^5 - 1/2*a^3, 1/4*a^20 - 1/4*a^18 - 1/2*a^16 + 1/4*a^14 - 1/2*a^13 + 1/4*a^12 - 1/2*a^11 - 1/4*a^10 - 1/2*a^9 - 1/2*a^8 - 1/2*a^7 + 1/4*a^6 + 1/4*a^4 - 1/2*a^3, 1/4*a^21 - 1/4*a^19 - 1/2*a^17 + 1/4*a^15 - 1/2*a^14 + 1/4*a^13 - 1/2*a^12 - 1/4*a^11 - 1/2*a^10 - 1/2*a^9 - 1/2*a^8 + 1/4*a^7 + 1/4*a^5 - 1/2*a^4, 1/8*a^22 - 1/8*a^20 - 1/4*a^18 - 3/8*a^16 - 1/4*a^15 + 1/8*a^14 - 1/4*a^13 - 1/8*a^12 + 1/4*a^11 + 1/4*a^10 - 1/4*a^9 - 3/8*a^8 - 1/2*a^7 + 1/8*a^6 - 1/4*a^5 - 1/2*a^4, 1/8*a^23 - 1/8*a^21 - 1/4*a^19 - 3/8*a^17 - 1/4*a^16 + 1/8*a^15 - 1/4*a^14 - 1/8*a^13 + 1/4*a^12 + 1/4*a^11 - 1/4*a^10 - 3/8*a^9 - 1/2*a^8 + 1/8*a^7 - 1/4*a^6 - 1/2*a^5, 1/336*a^24 + 1/56*a^23 + 1/336*a^22 - 17/168*a^21 + 1/21*a^19 - 19/336*a^18 - 17/84*a^17 - 113/336*a^16 + 11/84*a^15 + 1/336*a^14 + 5/28*a^13 + 5/42*a^12 - 1/24*a^11 + 27/112*a^10 - 1/168*a^9 - 15/112*a^8 - 1/21*a^7 + 3/8*a^6 - 3/14*a^5 - 13/84*a^4 + 3/7*a^3 - 4/21*a + 1/21, 1/336*a^25 + 1/48*a^23 + 1/168*a^22 - 1/56*a^21 - 13/168*a^20 - 31/336*a^19 - 19/168*a^18 - 167/336*a^17 - 10/21*a^16 - 137/336*a^15 + 1/28*a^14 + 29/168*a^13 + 31/84*a^12 + 55/112*a^11 - 19/42*a^10 + 31/112*a^9 - 5/42*a^8 - 3/14*a^7 + 23/56*a^6 - 5/42*a^5 - 1/7*a^4 + 3/7*a^3 - 4/21*a^2 + 4/21*a - 2/7, 1/8555232*a^26 - 379/1069404*a^25 + 9139/8555232*a^24 - 19281/356468*a^23 + 202837/4277616*a^22 - 15367/1069404*a^21 - 244801/2851744*a^20 - 166055/1425872*a^19 + 270317/8555232*a^18 - 659415/1425872*a^17 - 425519/2851744*a^16 - 1110559/4277616*a^15 - 808657/4277616*a^14 - 933689/4277616*a^13 - 981059/8555232*a^12 - 395105/2138808*a^11 - 3022619/8555232*a^10 - 421313/4277616*a^9 + 230263/1069404*a^8 - 13919/89117*a^7 + 448585/1069404*a^6 - 10763/267351*a^5 - 1343/89117*a^4 + 5230/267351*a^3 + 7930/89117*a^2 + 55931/267351*a + 627/89117, 1/8555232*a^27 + 7897/8555232*a^25 + 197/712936*a^24 - 26207/712936*a^23 - 4277/152772*a^22 - 22819/295008*a^21 - 323467/4277616*a^20 - 681733/8555232*a^19 + 73859/4277616*a^18 + 2402585/8555232*a^17 - 2048293/4277616*a^16 - 37648/89117*a^15 - 1885979/4277616*a^14 + 213835/2851744*a^13 - 598351/2138808*a^12 + 1293571/8555232*a^11 - 341039/4277616*a^10 - 535489/4277616*a^9 - 165115/712936*a^8 + 54245/267351*a^7 - 308983/712936*a^6 + 24565/50924*a^5 + 187627/534702*a^4 + 23917/534702*a^3 + 22032/89117*a^2 - 30440/267351*a + 63359/267351, 1/17880434880*a^28 + 211/8940217440*a^27 - 11/232213440*a^26 + 5553337/8940217440*a^25 - 614671/447010872*a^24 - 6246727/127717392*a^23 - 187532801/5960144960*a^22 - 4820043/78422960*a^21 + 673921309/5960144960*a^20 + 12089465/223505436*a^19 - 863782915/3576086976*a^18 + 414498767/1117527180*a^17 - 22992538/55876359*a^16 - 55063451/812747040*a^15 + 285526855/1192028992*a^14 + 260431565/596014496*a^13 + 2250470049/5960144960*a^12 + 736765027/1490036240*a^11 - 902218109/8940217440*a^10 - 426886445/894021744*a^9 - 404586643/894021744*a^8 - 9824039/50796690*a^7 + 1022203/319293480*a^6 - 40143434/279381795*a^5 - 9155921/38535420*a^4 - 23811454/55876359*a^3 + 81393091/186254530*a^2 + 3394115/55876359*a + 1992179/13303895, 1/5164304785068194153704857673809600*a^29 + 217390513579358576101/12719962524798507767745954861600*a^28 + 17582498244626722532863991/469482253188017650336805243073600*a^27 + 3867198668956995691626611/92219728304789181316158172746600*a^26 - 308902194111935534268041650849/322769049066762134606553604613100*a^25 - 199771164436298126418223552881/172143492835606471790161922460320*a^24 + 8606529593702409172255431865937/5164304785068194153704857673809600*a^23 - 32622330509970543757646149868669/1291076196267048538426214418452400*a^22 + 176163506419981961875743856364891/5164304785068194153704857673809600*a^21 + 198046479952147168461808180418959/2582152392534097076852428836904800*a^20 - 36934351765966778360282204523187/206572191402727766148194306952384*a^19 - 404298993033349484920842746559259/2582152392534097076852428836904800*a^18 + 327622822941902474552756057435137/1291076196267048538426214418452400*a^17 - 492906075824054331082793925857/4047260803344979744282803819600*a^16 + 793500155621469910210298010522379/1721434928356064717901619224603200*a^15 + 4182149417962481492054235250923/24591927547943781684308846065760*a^14 + 1249669835516787666269369556330227/5164304785068194153704857673809600*a^13 - 384973277261333614530212209990133/860717464178032358950809612301600*a^12 + 98344128572847908861802989717357/368878913219156725264632690986400*a^11 + 5924054820121194613202992057707/45300919167264860997411032226400*a^10 - 80988722273090477814878907246017/258215239253409707685242883690480*a^9 - 24588089729953557841456144033877/117370563297004412584201310768400*a^8 - 144331487968091020668134706775969/322769049066762134606553604613100*a^7 - 7066031732602085725254069017027/46109864152394590658079086373300*a^6 - 3044113478189382249387887746994/26897420755563511217212800384425*a^5 + 7392051960849255907983282379659/53794841511127022434425600768850*a^4 - 12057406529751270917400058732126/80692262266690533651638401153275*a^3 - 62415782607326988065034080732503/161384524533381067303276802306550*a^2 - 1898902722207843356680884377116/80692262266690533651638401153275*a - 3095565270019416682845364832224/7335660206062775786512581923025, 1/10328609570136388307409715347619200*a^30 - 15735235634367839771133/688573971342425887160647689841280*a^28 - 67690622543430729567523/9221972830478918131615817274660*a^27 + 1269459000840261558507937/86071746417803235895080961230160*a^26 - 1439249982116773704878253596927/1291076196267048538426214418452400*a^25 - 6958027411826394797692149230983/10328609570136388307409715347619200*a^24 - 297824346386994583033061955346049/5164304785068194153704857673809600*a^23 - 249754302230711516395648330260473/10328609570136388307409715347619200*a^22 + 91686399504441243968602862591527/1721434928356064717901619224603200*a^21 + 1254753706502091528084033710588597/10328609570136388307409715347619200*a^20 - 1154802358629236196594444729909709/5164304785068194153704857673809600*a^19 - 140400939248140003956098363677411/2582152392534097076852428836904800*a^18 - 156755767118996836641404999556259/1032860957013638830740971534761920*a^17 - 63982370333220117806278890795323/134137786625147900096230069449600*a^16 + 178305968734991029143104878375183/645538098133524269213107209226200*a^15 - 1379799020784523490661976060118651/3442869856712129435803238449206400*a^14 + 14255702558003599324544651281369/93896450637603530067361048614720*a^13 - 1080915860344578017137088164287467/5164304785068194153704857673809600*a^12 - 16387564850971771030456377150349/516430478506819415370485767380960*a^11 + 249214464504423684780530513727947/645538098133524269213107209226200*a^10 - 55853447634955853684994192615557/161384524533381067303276802306550*a^9 + 194556913528337629286508101571203/1291076196267048538426214418452400*a^8 - 1856470877347224513807572764973/3912352109900147086140043692280*a^7 - 74396762648690532731395699833283/215179366044508089737702403075400*a^6 + 6468339745811252014294629291554/80692262266690533651638401153275*a^5 - 26694554743391167898632008964349/53794841511127022434425600768850*a^4 + 14923756790273152319254737981271/80692262266690533651638401153275*a^3 - 19125995997203608158910625030447/161384524533381067303276802306550*a^2 - 30242704260507092349199168786909/80692262266690533651638401153275*a - 1620622703222053743442046392304/11527466038098647664519771593325, 1/10328609570136388307409715347619200*a^31 - 1/10328609570136388307409715347619200*a^29 + 7559610264912318967991/737757826438313450529265381972800*a^28 - 96569832611092112705713463/5164304785068194153704857673809600*a^27 + 57350619873850226086142033/1721434928356064717901619224603200*a^26 - 838918830143111276173240531457/1475515652876626901058530763945600*a^25 - 570956652618788414280982189469/469482253188017650336805243073600*a^24 - 67451040630618985256823287025979/2065721914027277661481943069523840*a^23 - 53817872136831151726949639860549/1291076196267048538426214418452400*a^22 + 962421559853171879601017240244671/10328609570136388307409715347619200*a^21 + 45181754695961994771870987963913/368878913219156725264632690986400*a^20 + 90188044945711713168772922773253/5164304785068194153704857673809600*a^19 - 34262463546588843505554890669369/368878913219156725264632690986400*a^18 + 1696538717837690291757680407558927/3442869856712129435803238449206400*a^17 + 4319067537519614590121468461409/129107619626704853842621441845240*a^16 + 711531935479156496846926189375097/2065721914027277661481943069523840*a^15 + 48411305051433706037515609898957/129107619626704853842621441845240*a^14 + 101652379568619004783749985135923/215179366044508089737702403075400*a^13 + 76891675223615168086957708204559/469482253188017650336805243073600*a^12 + 100931159636003975759500658616353/215179366044508089737702403075400*a^11 - 111077123801655521334923297225237/1291076196267048538426214418452400*a^10 - 14839078300784308366605579392289/430358732089016179475404806150800*a^9 + 146368952543198176046522821238117/430358732089016179475404806150800*a^8 - 50608290190599381727229109653449/129107619626704853842621441845240*a^7 - 70047011994711249736853170732237/322769049066762134606553604613100*a^6 - 4536987200486520347357480571301/16138452453338106730327680230655*a^5 - 4102270825181330977882932946387/23054932076197295329039543186650*a^4 + 2465791340650367923762906920553/53794841511127022434425600768850*a^3 - 916534184908881972873066896513/2782491802299673574194427625975*a^2 + 191163811960891399103146059412/2305493207619729532903954318665*a - 18400671885311529434010754470443/80692262266690533651638401153275

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

oscar: basis(OK)

Monogenic:		No
Index:		Not computed
Inessential primes:		$2$

Class group and class number

not computed

sage: K.class_group().invariants()

gp: K.clgp

magma: ClassGroup(K);

oscar: class_group(K)

Unit group

sage: UK = K.unit_group()

magma: UK, fUK := UnitGroup(K);

oscar: UK, fUK = unit_group(OK)

Rank:	$15$	sage: UK.rank() gp: K.fu magma: UnitRank(K); oscar: rank(UK)
Torsion generator:	\( \frac{82175595185965938672707651061}{11325229791816215249352758056600} a^{31} - \frac{26701819999859913810235209629}{1698784468772432287402913708490} a^{30} + \frac{13706791363226770543279549359}{1415653723977026906169094757075} a^{29} - \frac{1671152306197306775677274851841}{22650459583632430498705516113200} a^{28} - \frac{1506943496369743649854791473343}{11325229791816215249352758056600} a^{27} - \frac{16356534095865417746439594112931}{16987844687724322874029137084900} a^{26} + \frac{18015989745671620884900856714651}{4853669910778377964008324881400} a^{25} + \frac{40725804854207092887382951058213}{16987844687724322874029137084900} a^{24} + \frac{113991246225722764379634139412731}{11325229791816215249352758056600} a^{23} - \frac{5097729093982118295010770760363}{1132522979181621524935275805660} a^{22} + \frac{61140042844982344495276602880473}{404472492564864830334027073450} a^{21} - \frac{14356461068927968888762883763803543}{16987844687724322874029137084900} a^{20} + \frac{10174948495183650040658189021999363}{33975689375448645748058274169800} a^{19} - \frac{37244919594318353754509517131809061}{67951378750897291496116548339600} a^{18} + \frac{1280019767873655751586519199830651}{617739806826339013601059530360} a^{17} - \frac{22000380562174585061581977727602199}{1544349517065847534002648825900} a^{16} + \frac{1126630771778838593691159169621089799}{16987844687724322874029137084900} a^{15} - \frac{4635542506904882232781517606789149}{77217475853292376700132441295} a^{14} - \frac{23486210697672924247031233563400291}{1698784468772432287402913708490} a^{13} + \frac{460102146078922789207275207650823929}{4246961171931080718507284271225} a^{12} - \frac{574791037378212255886983800795259817}{4246961171931080718507284271225} a^{11} + \frac{76278049075806349923722190381186046}{1415653723977026906169094757075} a^{10} + \frac{16283810845399857035078901961888778}{128695793088820627833554068825} a^{9} - \frac{2540981192431039565516625508570012423}{9707339821556755928016649762800} a^{8} + \frac{793366949813010457078359422801356028}{4246961171931080718507284271225} a^{7} + \frac{183857073164236699454522670426163424}{4246961171931080718507284271225} a^{6} + \frac{28247335603402843542673245984163616}{4246961171931080718507284271225} a^{5} + \frac{1523561486501767148210046857652176}{4246961171931080718507284271225} a^{4} - \frac{225468321952735216276132948492996}{4246961171931080718507284271225} a^{3} + \frac{172658858323415880000941410095072}{1415653723977026906169094757075} a^{2} + \frac{37248650865183486278795687381536}{1415653723977026906169094757075} a + \frac{15118514187288634740492822019328}{4246961171931080718507284271225} \) (82175595185965938672707651061)/(11325229791816215249352758056600)a^(31) - (26701819999859913810235209629)/(1698784468772432287402913708490)a^(30) + (13706791363226770543279549359)/(1415653723977026906169094757075)a^(29) - (1671152306197306775677274851841)/(22650459583632430498705516113200)a^(28) - (1506943496369743649854791473343)/(11325229791816215249352758056600)a^(27) - (16356534095865417746439594112931)/(16987844687724322874029137084900)a^(26) + (18015989745671620884900856714651)/(4853669910778377964008324881400)a^(25) + (40725804854207092887382951058213)/(16987844687724322874029137084900)a^(24) + (113991246225722764379634139412731)/(11325229791816215249352758056600)a^(23) - (5097729093982118295010770760363)/(1132522979181621524935275805660)a^(22) + (61140042844982344495276602880473)/(404472492564864830334027073450)a^(21) - (14356461068927968888762883763803543)/(16987844687724322874029137084900)a^(20) + (10174948495183650040658189021999363)/(33975689375448645748058274169800)a^(19) - (37244919594318353754509517131809061)/(67951378750897291496116548339600)a^(18) + (1280019767873655751586519199830651)/(617739806826339013601059530360)a^(17) - (22000380562174585061581977727602199)/(1544349517065847534002648825900)a^(16) + (1126630771778838593691159169621089799)/(16987844687724322874029137084900)a^(15) - (4635542506904882232781517606789149)/(77217475853292376700132441295)a^(14) - (23486210697672924247031233563400291)/(1698784468772432287402913708490)a^(13) + (460102146078922789207275207650823929)/(4246961171931080718507284271225)a^(12) - (574791037378212255886983800795259817)/(4246961171931080718507284271225)a^(11) + (76278049075806349923722190381186046)/(1415653723977026906169094757075)a^(10) + (16283810845399857035078901961888778)/(128695793088820627833554068825)a^(9) - (2540981192431039565516625508570012423)/(9707339821556755928016649762800)a^(8) + (793366949813010457078359422801356028)/(4246961171931080718507284271225)a^(7) + (183857073164236699454522670426163424)/(4246961171931080718507284271225)a^(6) + (28247335603402843542673245984163616)/(4246961171931080718507284271225)a^(5) + (1523561486501767148210046857652176)/(4246961171931080718507284271225)a^(4) - (225468321952735216276132948492996)/(4246961171931080718507284271225)a^(3) + (172658858323415880000941410095072)/(1415653723977026906169094757075)a^(2) + (37248650865183486278795687381536)/(1415653723977026906169094757075)*a + (15118514187288634740492822019328)/(4246961171931080718507284271225) (order $30$)	sage: UK.torsion_generator() gp: K.tu[2] magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); oscar: torsion_units_generator(OK)
Fundamental units:	not computed	sage: UK.fundamental_units() gp: K.fu magma: [K\|fUK(g): g in Generators(UK)]; oscar: [K(fUK(a)) for a in gens(UK)]
Regulator:	not computed	sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr $ not computed \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula

x = polygen(QQ); K.<a> = NumberField(x^32 - 2*x^31 + x^30 - 10*x^29 - 20*x^28 - 136*x^27 + 489*x^26 + 412*x^25 + 1455*x^24 - 380*x^23 + 20765*x^22 - 113020*x^21 + 22444*x^20 - 71630*x^19 + 273805*x^18 - 1917530*x^17 + 8820755*x^16 - 6803856*x^15 - 3050102*x^14 + 14422848*x^13 - 16249472*x^12 + 4718368*x^11 + 18224552*x^10 - 33044576*x^9 + 20240880*x^8 + 9340480*x^7 + 2473504*x^6 + 461696*x^5 - 5632*x^4 + 15872*x^3 + 6272*x^2 + 1536*x + 256)

DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()

hK = K.class_number(); wK = K.unit_group().torsion_generator().order();

2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))

# self-contained Pari/GP code snippet to compute the analytic class number formula

K = bnfinit(x^32 - 2*x^31 + x^30 - 10*x^29 - 20*x^28 - 136*x^27 + 489*x^26 + 412*x^25 + 1455*x^24 - 380*x^23 + 20765*x^22 - 113020*x^21 + 22444*x^20 - 71630*x^19 + 273805*x^18 - 1917530*x^17 + 8820755*x^16 - 6803856*x^15 - 3050102*x^14 + 14422848*x^13 - 16249472*x^12 + 4718368*x^11 + 18224552*x^10 - 33044576*x^9 + 20240880*x^8 + 9340480*x^7 + 2473504*x^6 + 461696*x^5 - 5632*x^4 + 15872*x^3 + 6272*x^2 + 1536*x + 256, 1);

[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]

/* self-contained Magma code snippet to compute the analytic class number formula */

Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^32 - 2*x^31 + x^30 - 10*x^29 - 20*x^28 - 136*x^27 + 489*x^26 + 412*x^25 + 1455*x^24 - 380*x^23 + 20765*x^22 - 113020*x^21 + 22444*x^20 - 71630*x^19 + 273805*x^18 - 1917530*x^17 + 8820755*x^16 - 6803856*x^15 - 3050102*x^14 + 14422848*x^13 - 16249472*x^12 + 4718368*x^11 + 18224552*x^10 - 33044576*x^9 + 20240880*x^8 + 9340480*x^7 + 2473504*x^6 + 461696*x^5 - 5632*x^4 + 15872*x^3 + 6272*x^2 + 1536*x + 256);

OK := Integers(K); DK := Discriminant(OK);

UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);

r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);

hK := #clK; wK := #TorsionSubgroup(UK);

2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));

# self-contained Oscar code snippet to compute the analytic class number formula

Qx, x = PolynomialRing(QQ); K, a = NumberField(x^32 - 2*x^31 + x^30 - 10*x^29 - 20*x^28 - 136*x^27 + 489*x^26 + 412*x^25 + 1455*x^24 - 380*x^23 + 20765*x^22 - 113020*x^21 + 22444*x^20 - 71630*x^19 + 273805*x^18 - 1917530*x^17 + 8820755*x^16 - 6803856*x^15 - 3050102*x^14 + 14422848*x^13 - 16249472*x^12 + 4718368*x^11 + 18224552*x^10 - 33044576*x^9 + 20240880*x^8 + 9340480*x^7 + 2473504*x^6 + 461696*x^5 - 5632*x^4 + 15872*x^3 + 6272*x^2 + 1536*x + 256);

OK = ring_of_integers(K); DK = discriminant(OK);

UK, fUK = unit_group(OK); clK, fclK = class_group(OK);

r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);

hK = order(clK); wK = torsion_units_order(K);

2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))

Galois group

$C_2^3\times C_4$ (as 32T34):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: G = GaloisGroup(K);

oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)

An abelian group of order 32

The 32 conjugacy class representatives for $C_2^3\times C_4$

Character table for $C_2^3\times C_4$

Intermediate fields

$\Q(\sqrt{-35}) $, $\Q(\sqrt{2}) $, $\Q(\sqrt{-70}) $, $\Q(\sqrt{210}) $, $\Q(\sqrt{-6}) $, $\Q(\sqrt{105}) $, $\Q(\sqrt{-3}) $, $\Q(\sqrt{5}) $, $\Q(\sqrt{-7}) $, $\Q(\sqrt{10}) $, $\Q(\sqrt{-14}) $, $\Q(\sqrt{42}) $, $\Q(\sqrt{-30}) $, $\Q(\sqrt{21}) $, $\Q(\sqrt{-15}) $, $\Q(\sqrt{2}, \sqrt{-35})$, $\Q(\sqrt{-6}, \sqrt{-35})$, $\Q(\sqrt{-3}, \sqrt{-35})$, $\Q(\sqrt{2}, \sqrt{105})$, $\Q(\sqrt{2}, \sqrt{-3})$, $\Q(\sqrt{-3}, \sqrt{-70})$, $\Q(\sqrt{-6}, \sqrt{-70})$, $\Q(\sqrt{5}, \sqrt{-7})$, $\Q(\sqrt{10}, \sqrt{-14})$, $\Q(\sqrt{-30}, \sqrt{-35})$, $\Q(\sqrt{-15}, \sqrt{21})$, $\Q(\sqrt{2}, \sqrt{5})$, $\Q(\sqrt{2}, \sqrt{-7})$, $\Q(\sqrt{2}, \sqrt{21})$, $\Q(\sqrt{2}, \sqrt{-15})$, $\Q(\sqrt{5}, \sqrt{-14})$, $\Q(\sqrt{-7}, \sqrt{10})$, $\Q(\sqrt{-15}, \sqrt{42})$, $\Q(\sqrt{21}, \sqrt{-30})$, $\Q(\sqrt{5}, \sqrt{42})$, $\Q(\sqrt{-7}, \sqrt{-30})$, $\Q(\sqrt{10}, \sqrt{21})$, $\Q(\sqrt{-14}, \sqrt{-15})$, $\Q(\sqrt{5}, \sqrt{-6})$, $\Q(\sqrt{-6}, \sqrt{-7})$, $\Q(\sqrt{-6}, \sqrt{10})$, $\Q(\sqrt{-6}, \sqrt{-14})$, $\Q(\sqrt{5}, \sqrt{21})$, $\Q(\sqrt{-7}, \sqrt{-15})$, $\Q(\sqrt{10}, \sqrt{42})$, $\Q(\sqrt{-14}, \sqrt{-30})$, $\Q(\sqrt{-3}, \sqrt{5})$, $\Q(\sqrt{-3}, \sqrt{-7})$, $\Q(\sqrt{-3}, \sqrt{10})$, $\Q(\sqrt{-3}, \sqrt{-14})$, 4.4.6125.1, $\Q(\zeta_{5})$, 4.4.392000.1, 4.0.8000.2, 4.4.72000.1, 4.0.3528000.1, $\Q(\zeta_{15})^+$, 4.0.55125.1, 8.0.497871360000.14, 8.0.6146560000.2, 8.0.497871360000.1, 8.0.497871360000.9, 8.0.497871360000.5, 8.0.121550625.1, 8.0.497871360000.11, 8.8.497871360000.1, 8.0.497871360000.10, 8.0.207360000.1, 8.0.796594176.2, 8.0.497871360000.19, 8.0.497871360000.16, 8.0.497871360000.6, 8.0.497871360000.7, 8.0.37515625.1, 8.0.153664000000.2, 8.0.12446784000000.15, 8.0.3038765625.2, 8.8.153664000000.1, 8.0.64000000.2, 8.8.5184000000.1, 8.0.12446784000000.14, 8.0.153664000000.1, 8.0.153664000000.5, 8.0.12446784000000.3, 8.0.12446784000000.16, 8.8.12446784000000.4, 8.0.12446784000000.20, 8.8.12446784000000.6, 8.0.12446784000000.11, 8.0.12446784000000.6, 8.0.5184000000.5, 8.0.12446784000000.8, 8.0.5184000000.1, 8.8.3038765625.1, 8.0.3038765625.3, 8.8.12446784000000.3, 8.0.12446784000000.5, 8.0.3038765625.1, $\Q(\zeta_{15})$, 8.0.12446784000000.18, 8.0.5184000000.3, 16.0.247875891108249600000000.1, 16.0.23612624896000000000000.2, 16.0.154922431942656000000000000.13, 16.0.154922431942656000000000000.8, 16.0.154922431942656000000000000.18, 16.0.9234096523681640625.1, 16.0.154922431942656000000000000.14, 16.16.154922431942656000000000000.1, 16.0.154922431942656000000000000.20, 16.0.154922431942656000000000000.7, 16.0.26873856000000000000.2, 16.0.154922431942656000000000000.11, 16.0.154922431942656000000000000.2, 16.0.154922431942656000000000000.4, 16.0.154922431942656000000000000.5

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

sage: K.subfields()[1:-1]

gp: L = nfsubfields(K); L[2..length(b)]

magma: L := Subfields(K); L[2..#L];

oscar: subfields(K)[2:end-1]

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	R	R	R	R	${\href{/padicField/11.2.0.1}{2} }^{16}$	${\href{/padicField/13.4.0.1}{4} }^{8}$	${\href{/padicField/17.4.0.1}{4} }^{8}$	${\href{/padicField/19.2.0.1}{2} }^{16}$	${\href{/padicField/23.4.0.1}{4} }^{8}$	${\href{/padicField/29.2.0.1}{2} }^{16}$	${\href{/padicField/31.2.0.1}{2} }^{16}$	${\href{/padicField/37.4.0.1}{4} }^{8}$	${\href{/padicField/41.2.0.1}{2} }^{16}$	${\href{/padicField/43.4.0.1}{4} }^{8}$	${\href{/padicField/47.4.0.1}{4} }^{8}$	${\href{/padicField/53.4.0.1}{4} }^{8}$	${\href{/padicField/59.2.0.1}{2} }^{16}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:

p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]

\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:

p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])

// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:

p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:

p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]

Local algebras for ramified primes

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
$2$ 2	2.8.12.1	$x^{8} - 12 x^{7} + 52 x^{6} + 840 x^{5} + 3808 x^{4} + 10224 x^{3} + 17968 x^{2} + 20576 x + 15216$	$2$	$4$	$12$	$C_4\times C_2$	$[3]^{4}$
	2.8.12.1	$x^{8} - 12 x^{7} + 52 x^{6} + 840 x^{5} + 3808 x^{4} + 10224 x^{3} + 17968 x^{2} + 20576 x + 15216$	$2$	$4$	$12$	$C_4\times C_2$	$[3]^{4}$
	2.8.12.1	$x^{8} - 12 x^{7} + 52 x^{6} + 840 x^{5} + 3808 x^{4} + 10224 x^{3} + 17968 x^{2} + 20576 x + 15216$	$2$	$4$	$12$	$C_4\times C_2$	$[3]^{4}$
	2.8.12.1	$x^{8} - 12 x^{7} + 52 x^{6} + 840 x^{5} + 3808 x^{4} + 10224 x^{3} + 17968 x^{2} + 20576 x + 15216$	$2$	$4$	$12$	$C_4\times C_2$	$[3]^{4}$
$3$ 3	3.8.4.1	$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$	$2$	$4$	$4$	$C_4\times C_2$	$[\ ]_{2}^{4}$
	3.8.4.1	$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$	$2$	$4$	$4$	$C_4\times C_2$	$[\ ]_{2}^{4}$
	3.8.4.1	$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$	$2$	$4$	$4$	$C_4\times C_2$	$[\ ]_{2}^{4}$
	3.8.4.1	$x^{8} + 4 x^{7} + 16 x^{6} + 36 x^{5} + 94 x^{4} + 116 x^{3} + 144 x^{2} + 36 x + 229$	$2$	$4$	$4$	$C_4\times C_2$	$[\ ]_{2}^{4}$
$5$ 5	5.8.6.1	$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$	$4$	$2$	$6$	$C_4\times C_2$	$[\ ]_{4}^{2}$
	5.8.6.1	$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$	$4$	$2$	$6$	$C_4\times C_2$	$[\ ]_{4}^{2}$
	5.8.6.1	$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$	$4$	$2$	$6$	$C_4\times C_2$	$[\ ]_{4}^{2}$
	5.8.6.1	$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$	$4$	$2$	$6$	$C_4\times C_2$	$[\ ]_{4}^{2}$
$7$ 7	7.8.4.1	$x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$	$2$	$4$	$4$	$C_4\times C_2$	$[\ ]_{2}^{4}$
	7.8.4.1	$x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$	$2$	$4$	$4$	$C_4\times C_2$	$[\ ]_{2}^{4}$
	7.8.4.1	$x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$	$2$	$4$	$4$	$C_4\times C_2$	$[\ ]_{2}^{4}$
	7.8.4.1	$x^{8} + 38 x^{6} + 8 x^{5} + 395 x^{4} - 72 x^{3} + 1026 x^{2} - 872 x + 401$	$2$	$4$	$4$	$C_4\times C_2$	$[\ ]_{2}^{4}$

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