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Properties

Label 32.0.240...000.1
Degree $32$
Signature $[0, 16]$
Discriminant $2.400\times 10^{52}$
Root discriminant \(43.34\)
Ramified primes $2,3,5,7$
Class number not computed
Class group not computed
Galois group $C_2^3\times C_4$ (as 32T34)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 2*x^31 + 9*x^30 - 30*x^29 + 46*x^28 + 64*x^27 - 323*x^26 + 2264*x^25 - 7635*x^24 + 21504*x^23 - 30305*x^22 + 57880*x^21 + 39466*x^20 - 694266*x^19 + 2319339*x^18 - 3485698*x^17 + 14846451*x^16 - 12810752*x^15 + 2519526*x^14 + 3448824*x^13 - 3068744*x^12 + 1256816*x^11 - 39944*x^10 - 329952*x^9 + 240720*x^8 - 70592*x^7 - 12128*x^6 + 24320*x^5 - 13184*x^4 + 3072*x^3 + 1152*x^2 - 1024*x + 256)
 
gp: K = bnfinit(y^32 - 2*y^31 + 9*y^30 - 30*y^29 + 46*y^28 + 64*y^27 - 323*y^26 + 2264*y^25 - 7635*y^24 + 21504*y^23 - 30305*y^22 + 57880*y^21 + 39466*y^20 - 694266*y^19 + 2319339*y^18 - 3485698*y^17 + 14846451*y^16 - 12810752*y^15 + 2519526*y^14 + 3448824*y^13 - 3068744*y^12 + 1256816*y^11 - 39944*y^10 - 329952*y^9 + 240720*y^8 - 70592*y^7 - 12128*y^6 + 24320*y^5 - 13184*y^4 + 3072*y^3 + 1152*y^2 - 1024*y + 256, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 - 2*x^31 + 9*x^30 - 30*x^29 + 46*x^28 + 64*x^27 - 323*x^26 + 2264*x^25 - 7635*x^24 + 21504*x^23 - 30305*x^22 + 57880*x^21 + 39466*x^20 - 694266*x^19 + 2319339*x^18 - 3485698*x^17 + 14846451*x^16 - 12810752*x^15 + 2519526*x^14 + 3448824*x^13 - 3068744*x^12 + 1256816*x^11 - 39944*x^10 - 329952*x^9 + 240720*x^8 - 70592*x^7 - 12128*x^6 + 24320*x^5 - 13184*x^4 + 3072*x^3 + 1152*x^2 - 1024*x + 256);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^32 - 2*x^31 + 9*x^30 - 30*x^29 + 46*x^28 + 64*x^27 - 323*x^26 + 2264*x^25 - 7635*x^24 + 21504*x^23 - 30305*x^22 + 57880*x^21 + 39466*x^20 - 694266*x^19 + 2319339*x^18 - 3485698*x^17 + 14846451*x^16 - 12810752*x^15 + 2519526*x^14 + 3448824*x^13 - 3068744*x^12 + 1256816*x^11 - 39944*x^10 - 329952*x^9 + 240720*x^8 - 70592*x^7 - 12128*x^6 + 24320*x^5 - 13184*x^4 + 3072*x^3 + 1152*x^2 - 1024*x + 256)
 

\( x^{32} - 2 x^{31} + 9 x^{30} - 30 x^{29} + 46 x^{28} + 64 x^{27} - 323 x^{26} + 2264 x^{25} - 7635 x^{24} + \cdots + 256 \) Copy content Toggle raw display

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:   \(24000959919026880122072334336000000000000000000000000\) \(\medspace = 2^{48}\cdot 3^{16}\cdot 5^{24}\cdot 7^{16}\) Copy content Toggle raw display
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:   \(2\), \(3\), \(5\), \(7\) Copy content Toggle raw display
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$$\chi_{840}(1,·)$, $\chi_{840}(643,·)$, $\chi_{840}(769,·)$, $\chi_{840}(139,·)$, $\chi_{840}(659,·)$, $\chi_{840}(281,·)$, $\chi_{840}(673,·)$, $\chi_{840}(547,·)$, $\chi_{840}(41,·)$, $\chi_{840}(43,·)$, $\chi_{840}(433,·)$, $\chi_{840}(307,·)$, $\chi_{840}(827,·)$, $\chi_{840}(449,·)$, $\chi_{840}(83,·)$, $\chi_{840}(323,·)$, $\chi_{840}(713,·)$, $\chi_{840}(587,·)$, $\chi_{840}(811,·)$, $\chi_{840}(337,·)$, $\chi_{840}(419,·)$, $\chi_{840}(601,·)$, $\chi_{840}(97,·)$, $\chi_{840}(379,·)$, $\chi_{840}(209,·)$, $\chi_{840}(617,·)$, $\chi_{840}(491,·)$, $\chi_{840}(113,·)$, $\chi_{840}(211,·)$, $\chi_{840}(169,·)$, $\chi_{840}(377,·)$, $\chi_{840}(251,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{32768}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{22}a^{18}-\frac{2}{11}a^{17}-\frac{1}{2}a^{16}-\frac{1}{11}a^{15}+\frac{4}{11}a^{14}-\frac{7}{22}a^{12}+\frac{3}{11}a^{11}-\frac{1}{2}a^{10}-\frac{4}{11}a^{9}-\frac{1}{2}a^{8}-\frac{2}{11}a^{7}-\frac{3}{11}a^{6}-\frac{3}{22}a^{4}-\frac{5}{11}a^{3}-\frac{1}{2}a^{2}+\frac{3}{11}a-\frac{1}{11}$, $\frac{1}{22}a^{19}-\frac{5}{22}a^{17}-\frac{1}{11}a^{16}+\frac{5}{11}a^{14}-\frac{7}{22}a^{13}-\frac{9}{22}a^{11}-\frac{4}{11}a^{10}+\frac{1}{22}a^{9}-\frac{2}{11}a^{8}-\frac{1}{11}a^{6}-\frac{3}{22}a^{5}-\frac{7}{22}a^{3}+\frac{3}{11}a^{2}-\frac{4}{11}$, $\frac{1}{44}a^{20}-\frac{1}{44}a^{18}+\frac{1}{11}a^{17}-\frac{5}{11}a^{15}-\frac{19}{44}a^{14}-\frac{1}{2}a^{13}+\frac{7}{44}a^{12}-\frac{3}{22}a^{11}+\frac{1}{44}a^{10}-\frac{7}{22}a^{9}-\frac{9}{22}a^{7}-\frac{5}{44}a^{6}-\frac{19}{44}a^{4}+\frac{5}{22}a^{3}+\frac{4}{11}a-\frac{2}{11}$, $\frac{1}{44}a^{21}-\frac{1}{44}a^{19}+\frac{4}{11}a^{17}-\frac{5}{11}a^{16}-\frac{1}{4}a^{15}-\frac{5}{22}a^{14}+\frac{7}{44}a^{13}-\frac{1}{2}a^{12}+\frac{21}{44}a^{11}-\frac{7}{22}a^{10}-\frac{3}{11}a^{9}-\frac{9}{22}a^{8}+\frac{1}{4}a^{7}-\frac{5}{11}a^{6}-\frac{19}{44}a^{5}-\frac{1}{2}a^{4}-\frac{1}{11}a^{3}+\frac{4}{11}a^{2}+\frac{3}{11}a+\frac{2}{11}$, $\frac{1}{88}a^{22}-\frac{1}{88}a^{20}-\frac{1}{8}a^{16}-\frac{1}{4}a^{15}-\frac{3}{8}a^{14}-\frac{1}{4}a^{13}-\frac{43}{88}a^{12}+\frac{1}{4}a^{11}-\frac{3}{22}a^{10}-\frac{1}{4}a^{9}-\frac{3}{8}a^{8}-\frac{1}{8}a^{6}-\frac{1}{4}a^{5}-\frac{1}{2}a^{3}-\frac{4}{11}a^{2}+\frac{4}{11}$, $\frac{1}{88}a^{23}-\frac{1}{88}a^{21}-\frac{1}{8}a^{17}-\frac{1}{4}a^{16}-\frac{3}{8}a^{15}-\frac{1}{4}a^{14}-\frac{43}{88}a^{13}+\frac{1}{4}a^{12}-\frac{3}{22}a^{11}-\frac{1}{4}a^{10}-\frac{3}{8}a^{9}-\frac{1}{8}a^{7}-\frac{1}{4}a^{6}-\frac{1}{2}a^{4}-\frac{4}{11}a^{3}+\frac{4}{11}a$, $\frac{1}{176}a^{24}-\frac{1}{176}a^{22}-\frac{3}{176}a^{18}+\frac{17}{88}a^{17}+\frac{5}{16}a^{16}-\frac{19}{88}a^{15}+\frac{21}{176}a^{14}-\frac{3}{8}a^{13}+\frac{5}{44}a^{12}-\frac{31}{88}a^{11}+\frac{5}{16}a^{10}+\frac{3}{22}a^{9}-\frac{1}{16}a^{8}-\frac{27}{88}a^{7}-\frac{3}{11}a^{6}-\frac{1}{4}a^{5}+\frac{2}{11}a^{4}+\frac{1}{22}a^{3}+\frac{2}{11}a^{2}+\frac{3}{11}a-\frac{1}{11}$, $\frac{1}{176}a^{25}-\frac{1}{176}a^{23}-\frac{3}{176}a^{19}+\frac{1}{88}a^{18}+\frac{7}{176}a^{17}-\frac{19}{88}a^{16}+\frac{85}{176}a^{15}+\frac{15}{88}a^{14}+\frac{5}{44}a^{13}-\frac{7}{88}a^{12}+\frac{39}{176}a^{11}+\frac{3}{22}a^{10}+\frac{69}{176}a^{9}-\frac{27}{88}a^{8}+\frac{5}{11}a^{7}-\frac{7}{44}a^{6}+\frac{2}{11}a^{5}-\frac{9}{22}a^{4}+\frac{3}{11}a^{2}-\frac{2}{11}a+\frac{4}{11}$, $\frac{1}{10038688}a^{26}-\frac{4391}{2509672}a^{25}-\frac{21001}{10038688}a^{24}-\frac{3379}{627418}a^{23}+\frac{717}{627418}a^{22}-\frac{21645}{2509672}a^{21}+\frac{49}{11552}a^{20}+\frac{100235}{5019344}a^{19}-\frac{65961}{10038688}a^{18}-\frac{1980819}{5019344}a^{17}-\frac{243249}{912608}a^{16}+\frac{1680295}{5019344}a^{15}+\frac{872673}{2509672}a^{14}+\frac{918419}{5019344}a^{13}-\frac{19073}{10038688}a^{12}-\frac{308227}{2509672}a^{11}+\frac{599397}{10038688}a^{10}+\frac{746813}{5019344}a^{9}-\frac{515419}{1254836}a^{8}-\frac{10189}{1254836}a^{7}+\frac{1092}{313709}a^{6}+\frac{211139}{1254836}a^{5}-\frac{219113}{627418}a^{4}+\frac{34467}{627418}a^{3}-\frac{146929}{313709}a^{2}-\frac{80392}{313709}a-\frac{71352}{313709}$, $\frac{1}{10038688}a^{27}+\frac{3445}{10038688}a^{25}+\frac{81}{132088}a^{24}-\frac{1}{176}a^{23}-\frac{12647}{2509672}a^{22}-\frac{15459}{10038688}a^{21}-\frac{8627}{5019344}a^{20}+\frac{204493}{10038688}a^{19}-\frac{1141}{63536}a^{18}-\frac{4514561}{10038688}a^{17}+\frac{1114143}{5019344}a^{16}+\frac{25111}{456304}a^{15}+\frac{2072109}{5019344}a^{14}+\frac{4248583}{10038688}a^{13}+\frac{1128483}{2509672}a^{12}-\frac{1488193}{10038688}a^{11}+\frac{1869083}{5019344}a^{10}-\frac{1810333}{5019344}a^{9}-\frac{229265}{2509672}a^{8}+\frac{1497}{1254836}a^{7}-\frac{412837}{1254836}a^{6}+\frac{622727}{1254836}a^{5}-\frac{367405}{1254836}a^{4}+\frac{282097}{627418}a^{3}+\frac{2}{11}a^{2}-\frac{15193}{313709}a+\frac{97965}{313709}$, $\frac{1}{2145568786240}a^{28}-\frac{47423}{1072784393120}a^{27}+\frac{6647}{195051707840}a^{26}-\frac{733248693}{1072784393120}a^{25}-\frac{431672017}{1072784393120}a^{24}+\frac{1415325663}{268196098280}a^{23}-\frac{4738068503}{2145568786240}a^{22}-\frac{4536571549}{536392196560}a^{21}+\frac{2296341461}{429113757248}a^{20}-\frac{10586013893}{536392196560}a^{19}+\frac{3897929099}{429113757248}a^{18}+\frac{129515086797}{536392196560}a^{17}-\frac{144038372319}{1072784393120}a^{16}+\frac{24065099643}{214556878624}a^{15}+\frac{10989655915}{429113757248}a^{14}+\frac{222626964093}{1072784393120}a^{13}+\frac{877632991}{14797026112}a^{12}+\frac{142164445653}{536392196560}a^{11}-\frac{40760444473}{1072784393120}a^{10}+\frac{577877629}{1849628264}a^{9}-\frac{27928474489}{536392196560}a^{8}+\frac{49250578349}{268196098280}a^{7}-\frac{492951875}{1849628264}a^{6}+\frac{3941796687}{13409804914}a^{5}+\frac{28071563383}{134098049140}a^{4}-\frac{28213028053}{67049024570}a^{3}+\frac{628528287}{3047682935}a^{2}+\frac{2477532483}{33524512285}a+\frac{16221056974}{33524512285}$, $\frac{1}{10\!\cdots\!00}a^{29}-\frac{1110104364491}{20\!\cdots\!40}a^{28}-\frac{31\!\cdots\!39}{10\!\cdots\!00}a^{27}-\frac{40\!\cdots\!49}{10\!\cdots\!00}a^{26}-\frac{45\!\cdots\!85}{20\!\cdots\!64}a^{25}+\frac{89\!\cdots\!73}{25\!\cdots\!80}a^{24}-\frac{24\!\cdots\!19}{10\!\cdots\!00}a^{23}+\frac{21\!\cdots\!91}{10\!\cdots\!00}a^{22}-\frac{11\!\cdots\!61}{10\!\cdots\!00}a^{21}+\frac{14\!\cdots\!53}{10\!\cdots\!00}a^{20}+\frac{18\!\cdots\!53}{10\!\cdots\!00}a^{19}+\frac{58\!\cdots\!23}{10\!\cdots\!00}a^{18}+\frac{23\!\cdots\!39}{46\!\cdots\!60}a^{17}+\frac{65\!\cdots\!01}{51\!\cdots\!00}a^{16}+\frac{25\!\cdots\!89}{20\!\cdots\!40}a^{15}-\frac{48\!\cdots\!29}{10\!\cdots\!00}a^{14}+\frac{21\!\cdots\!91}{10\!\cdots\!00}a^{13}+\frac{38\!\cdots\!07}{10\!\cdots\!00}a^{12}+\frac{12\!\cdots\!69}{25\!\cdots\!00}a^{11}+\frac{20\!\cdots\!23}{12\!\cdots\!00}a^{10}+\frac{35\!\cdots\!73}{12\!\cdots\!00}a^{9}+\frac{68\!\cdots\!59}{25\!\cdots\!00}a^{8}+\frac{61\!\cdots\!93}{16\!\cdots\!75}a^{7}+\frac{17\!\cdots\!29}{64\!\cdots\!70}a^{6}-\frac{43\!\cdots\!01}{32\!\cdots\!50}a^{5}+\frac{65\!\cdots\!71}{32\!\cdots\!50}a^{4}+\frac{70\!\cdots\!88}{16\!\cdots\!75}a^{3}+\frac{57\!\cdots\!54}{16\!\cdots\!65}a^{2}-\frac{71\!\cdots\!23}{16\!\cdots\!75}a+\frac{23\!\cdots\!59}{16\!\cdots\!75}$, $\frac{1}{20\!\cdots\!00}a^{30}-\frac{48780641851}{10\!\cdots\!00}a^{28}-\frac{41\!\cdots\!91}{51\!\cdots\!00}a^{27}-\frac{35\!\cdots\!19}{93\!\cdots\!20}a^{26}-\frac{21\!\cdots\!49}{20\!\cdots\!64}a^{25}+\frac{56\!\cdots\!01}{20\!\cdots\!00}a^{24}-\frac{39\!\cdots\!87}{10\!\cdots\!00}a^{23}-\frac{18\!\cdots\!41}{20\!\cdots\!00}a^{22}+\frac{10\!\cdots\!39}{10\!\cdots\!00}a^{21}+\frac{35\!\cdots\!63}{18\!\cdots\!00}a^{20}-\frac{10\!\cdots\!51}{10\!\cdots\!00}a^{19}-\frac{19\!\cdots\!77}{51\!\cdots\!60}a^{18}+\frac{25\!\cdots\!39}{35\!\cdots\!00}a^{17}-\frac{21\!\cdots\!01}{41\!\cdots\!80}a^{16}-\frac{24\!\cdots\!01}{67\!\cdots\!00}a^{15}+\frac{75\!\cdots\!21}{20\!\cdots\!00}a^{14}-\frac{10\!\cdots\!09}{10\!\cdots\!00}a^{13}+\frac{13\!\cdots\!09}{51\!\cdots\!00}a^{12}-\frac{20\!\cdots\!37}{25\!\cdots\!00}a^{11}+\frac{21\!\cdots\!71}{51\!\cdots\!00}a^{10}-\frac{66\!\cdots\!73}{16\!\cdots\!75}a^{9}-\frac{11\!\cdots\!61}{25\!\cdots\!00}a^{8}-\frac{12\!\cdots\!37}{17\!\cdots\!80}a^{7}+\frac{34\!\cdots\!73}{12\!\cdots\!00}a^{6}+\frac{15\!\cdots\!54}{16\!\cdots\!75}a^{5}-\frac{70\!\cdots\!41}{16\!\cdots\!75}a^{4}+\frac{26\!\cdots\!03}{64\!\cdots\!70}a^{3}-\frac{60\!\cdots\!53}{29\!\cdots\!50}a^{2}+\frac{82\!\cdots\!27}{16\!\cdots\!75}a+\frac{16\!\cdots\!70}{64\!\cdots\!27}$, $\frac{1}{20\!\cdots\!00}a^{31}-\frac{1}{20\!\cdots\!00}a^{29}-\frac{3886571450711}{51\!\cdots\!00}a^{28}-\frac{17\!\cdots\!23}{51\!\cdots\!00}a^{27}-\frac{12\!\cdots\!53}{51\!\cdots\!00}a^{26}-\frac{51\!\cdots\!59}{20\!\cdots\!00}a^{25}+\frac{78\!\cdots\!73}{10\!\cdots\!00}a^{24}+\frac{22\!\cdots\!07}{20\!\cdots\!00}a^{23}-\frac{53\!\cdots\!57}{10\!\cdots\!00}a^{22}-\frac{54\!\cdots\!97}{21\!\cdots\!20}a^{21}-\frac{42\!\cdots\!99}{10\!\cdots\!00}a^{20}+\frac{10\!\cdots\!58}{13\!\cdots\!75}a^{19}-\frac{10\!\cdots\!23}{54\!\cdots\!00}a^{18}-\frac{16\!\cdots\!37}{41\!\cdots\!80}a^{17}-\frac{46\!\cdots\!81}{25\!\cdots\!00}a^{16}-\frac{30\!\cdots\!19}{20\!\cdots\!00}a^{15}+\frac{34\!\cdots\!51}{20\!\cdots\!40}a^{14}+\frac{78\!\cdots\!71}{51\!\cdots\!00}a^{13}-\frac{51\!\cdots\!73}{51\!\cdots\!60}a^{12}+\frac{25\!\cdots\!93}{51\!\cdots\!00}a^{11}+\frac{12\!\cdots\!51}{25\!\cdots\!00}a^{10}+\frac{20\!\cdots\!03}{25\!\cdots\!00}a^{9}-\frac{15\!\cdots\!21}{88\!\cdots\!00}a^{8}-\frac{46\!\cdots\!67}{16\!\cdots\!75}a^{7}+\frac{87\!\cdots\!83}{67\!\cdots\!00}a^{6}+\frac{30\!\cdots\!93}{64\!\cdots\!00}a^{5}-\frac{42\!\cdots\!77}{64\!\cdots\!00}a^{4}-\frac{36\!\cdots\!82}{16\!\cdots\!75}a^{3}+\frac{50\!\cdots\!32}{16\!\cdots\!75}a^{2}-\frac{64\!\cdots\!67}{16\!\cdots\!75}a+\frac{21\!\cdots\!76}{16\!\cdots\!75}$ Copy content Toggle raw display

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{14432947312558132091727}{4691564873085291653645600} a^{31} - \frac{19307346271358170092921}{2345782436542645826822800} a^{30} + \frac{139494822308011783945831}{4691564873085291653645600} a^{29} - \frac{50803868905985565896067}{469156487308529165364560} a^{28} + \frac{39954913720965059484231}{213252948776604166074800} a^{27} + \frac{44539963496654647744629}{293222804567830728352850} a^{26} - \frac{70238327210554363186707}{59386897127661919666400} a^{25} + \frac{1746680444103242578734191}{234578243654264582682280} a^{24} - \frac{25923774402203764655875833}{938312974617058330729120} a^{23} + \frac{91122198604939045833533973}{1172891218271322913411400} a^{22} - \frac{586770525084339713173901679}{4691564873085291653645600} a^{21} + \frac{241698080537910360827765793}{1172891218271322913411400} a^{20} + \frac{85041543571586112520863823}{2345782436542645826822800} a^{19} - \frac{5405533522880227905375842427}{2345782436542645826822800} a^{18} + \frac{71691515518362080064336171}{8514636793258242565600} a^{17} - \frac{1328988863626345784310034581}{93831297461705833072912} a^{16} + \frac{46019621238203788514934050673}{938312974617058330729120} a^{15} - \frac{77473816454016507091502712211}{1172891218271322913411400} a^{14} + \frac{16860145175685913630039798581}{2345782436542645826822800} a^{13} + \frac{6700146580655131879463744313}{586445609135661456705700} a^{12} - \frac{5784573597335708698417488687}{586445609135661456705700} a^{11} + \frac{46236412862086055387737779}{11728912182713229134114} a^{10} - \frac{11453110729489830740864107}{26656618597075520759350} a^{9} - \frac{31714172719880248539522927}{29322280456783072835285} a^{8} + \frac{225214149564969071349057219}{293222804567830728352850} a^{7} - \frac{2914283128240407776235786}{13328309298537760379675} a^{6} - \frac{330950399978600268757551}{7716389593890282325075} a^{5} + \frac{35056395949123325620556743}{586445609135661456705700} a^{4} - \frac{558360838402027954747308}{13328309298537760379675} a^{3} + \frac{1378754257745669516613648}{146611402283915364176425} a^{2} + \frac{10144769520475227253284}{2665661859707552075935} a - \frac{477947567707104796843968}{146611402283915364176425} \)  (order $30$) Copy content Toggle raw display
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 + 9*x^30 - 30*x^29 + 46*x^28 + 64*x^27 - 323*x^26 + 2264*x^25 - 7635*x^24 + 21504*x^23 - 30305*x^22 + 57880*x^21 + 39466*x^20 - 694266*x^19 + 2319339*x^18 - 3485698*x^17 + 14846451*x^16 - 12810752*x^15 + 2519526*x^14 + 3448824*x^13 - 3068744*x^12 + 1256816*x^11 - 39944*x^10 - 329952*x^9 + 240720*x^8 - 70592*x^7 - 12128*x^6 + 24320*x^5 - 13184*x^4 + 3072*x^3 + 1152*x^2 - 1024*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 + 9*x^30 - 30*x^29 + 46*x^28 + 64*x^27 - 323*x^26 + 2264*x^25 - 7635*x^24 + 21504*x^23 - 30305*x^22 + 57880*x^21 + 39466*x^20 - 694266*x^19 + 2319339*x^18 - 3485698*x^17 + 14846451*x^16 - 12810752*x^15 + 2519526*x^14 + 3448824*x^13 - 3068744*x^12 + 1256816*x^11 - 39944*x^10 - 329952*x^9 + 240720*x^8 - 70592*x^7 - 12128*x^6 + 24320*x^5 - 13184*x^4 + 3072*x^3 + 1152*x^2 - 1024*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 + 9*x^30 - 30*x^29 + 46*x^28 + 64*x^27 - 323*x^26 + 2264*x^25 - 7635*x^24 + 21504*x^23 - 30305*x^22 + 57880*x^21 + 39466*x^20 - 694266*x^19 + 2319339*x^18 - 3485698*x^17 + 14846451*x^16 - 12810752*x^15 + 2519526*x^14 + 3448824*x^13 - 3068744*x^12 + 1256816*x^11 - 39944*x^10 - 329952*x^9 + 240720*x^8 - 70592*x^7 - 12128*x^6 + 24320*x^5 - 13184*x^4 + 3072*x^3 + 1152*x^2 - 1024*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 + 9*x^30 - 30*x^29 + 46*x^28 + 64*x^27 - 323*x^26 + 2264*x^25 - 7635*x^24 + 21504*x^23 - 30305*x^22 + 57880*x^21 + 39466*x^20 - 694266*x^19 + 2319339*x^18 - 3485698*x^17 + 14846451*x^16 - 12810752*x^15 + 2519526*x^14 + 3448824*x^13 - 3068744*x^12 + 1256816*x^11 - 39944*x^10 - 329952*x^9 + 240720*x^8 - 70592*x^7 - 12128*x^6 + 24320*x^5 - 13184*x^4 + 3072*x^3 + 1152*x^2 - 1024*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 $C_2^3\times C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{105}) \), \(\Q(\sqrt{-210}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{70}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-35}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{-42}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{14}) \), \(\Q(\sqrt{30}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-2}, \sqrt{105})\), \(\Q(\sqrt{6}, \sqrt{70})\), \(\Q(\sqrt{-3}, \sqrt{-35})\), \(\Q(\sqrt{-3}, \sqrt{70})\), \(\Q(\sqrt{6}, \sqrt{-35})\), \(\Q(\sqrt{-2}, \sqrt{-35})\), \(\Q(\sqrt{-2}, \sqrt{-3})\), \(\Q(\sqrt{5}, \sqrt{21})\), \(\Q(\sqrt{-10}, \sqrt{-42})\), \(\Q(\sqrt{14}, \sqrt{30})\), \(\Q(\sqrt{-7}, \sqrt{-15})\), \(\Q(\sqrt{5}, \sqrt{-42})\), \(\Q(\sqrt{-10}, \sqrt{21})\), \(\Q(\sqrt{14}, \sqrt{-15})\), \(\Q(\sqrt{-7}, \sqrt{30})\), \(\Q(\sqrt{-2}, \sqrt{5})\), \(\Q(\sqrt{-2}, \sqrt{21})\), \(\Q(\sqrt{-2}, \sqrt{-7})\), \(\Q(\sqrt{-2}, \sqrt{-15})\), \(\Q(\sqrt{5}, \sqrt{14})\), \(\Q(\sqrt{21}, \sqrt{30})\), \(\Q(\sqrt{-15}, \sqrt{-42})\), \(\Q(\sqrt{-7}, \sqrt{-10})\), \(\Q(\sqrt{5}, \sqrt{6})\), \(\Q(\sqrt{6}, \sqrt{14})\), \(\Q(\sqrt{6}, \sqrt{-7})\), \(\Q(\sqrt{6}, \sqrt{-10})\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\sqrt{-3}, \sqrt{-7})\), \(\Q(\sqrt{-3}, \sqrt{14})\), \(\Q(\sqrt{-3}, \sqrt{-10})\), \(\Q(\sqrt{5}, \sqrt{-7})\), \(\Q(\sqrt{-15}, \sqrt{21})\), \(\Q(\sqrt{30}, \sqrt{-35})\), \(\Q(\sqrt{-10}, \sqrt{14})\), 4.4.3528000.1, 4.4.8000.1, \(\Q(\zeta_{5})\), 4.0.55125.1, \(\Q(\zeta_{15})^+\), 4.4.6125.1, 4.0.392000.2, 4.0.72000.2, 8.0.497871360000.17, 8.0.497871360000.13, 8.0.497871360000.3, 8.8.497871360000.2, 8.0.497871360000.2, 8.0.121550625.1, 8.0.497871360000.12, 8.0.497871360000.20, 8.0.497871360000.18, 8.0.497871360000.15, 8.0.497871360000.4, 8.0.6146560000.1, 8.0.497871360000.8, 8.0.207360000.2, 8.0.796594176.1, 8.8.12446784000000.2, 8.0.3038765625.3, 8.8.3038765625.1, 8.0.12446784000000.1, 8.0.12446784000000.19, 8.0.12446784000000.12, 8.0.12446784000000.13, 8.0.12446784000000.2, 8.0.12446784000000.7, 8.0.64000000.1, 8.0.5184000000.2, 8.0.153664000000.3, 8.8.12446784000000.5, 8.8.153664000000.2, 8.0.153664000000.6, 8.0.12446784000000.4, 8.8.12446784000000.1, 8.8.5184000000.2, 8.0.5184000000.6, 8.0.12446784000000.9, 8.0.12446784000000.17, 8.0.5184000000.4, \(\Q(\zeta_{15})\), 8.0.3038765625.1, 8.0.12446784000000.10, 8.0.153664000000.4, 8.0.37515625.1, 8.0.3038765625.2, 16.0.247875891108249600000000.2, 16.0.154922431942656000000000000.19, 16.0.154922431942656000000000000.9, 16.16.154922431942656000000000000.2, 16.0.154922431942656000000000000.1, 16.0.154922431942656000000000000.15, 16.0.9234096523681640625.1, 16.0.154922431942656000000000000.6, 16.0.154922431942656000000000000.3, 16.0.154922431942656000000000000.10, 16.0.154922431942656000000000000.17, 16.0.154922431942656000000000000.12, 16.0.23612624896000000000000.1, 16.0.154922431942656000000000000.16, 16.0.26873856000000000000.1

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display 2.8.12.2$x^{8} + 16 x^{7} + 120 x^{6} + 546 x^{5} + 1646 x^{4} + 3352 x^{3} + 4457 x^{2} + 3470 x + 1203$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.2$x^{8} + 16 x^{7} + 120 x^{6} + 546 x^{5} + 1646 x^{4} + 3352 x^{3} + 4457 x^{2} + 3470 x + 1203$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.2$x^{8} + 16 x^{7} + 120 x^{6} + 546 x^{5} + 1646 x^{4} + 3352 x^{3} + 4457 x^{2} + 3470 x + 1203$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
2.8.12.2$x^{8} + 16 x^{7} + 120 x^{6} + 546 x^{5} + 1646 x^{4} + 3352 x^{3} + 4457 x^{2} + 3470 x + 1203$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
\(3\) Copy content Toggle raw display 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}$
\(5\) Copy content Toggle raw display 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}$
\(7\) Copy content Toggle raw display 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}$