Properties

Label 32.0.503...000.1
Degree $32$
Signature $[0, 16]$
Discriminant $5.038\times 10^{48}$
Root discriminant \(33.26\)
Ramified primes $2,3,5,17$
Class number $16$ (GRH)
Class group [4, 4] (GRH)
Galois group $C_4^2:C_2^2$ (as 32T204)

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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 + 3*x^30 - 2*x^29 - 12*x^27 + 17*x^26 - 24*x^25 + 3*x^24 + 8*x^23 + 101*x^22 - 128*x^21 + 208*x^20 - 34*x^19 + 39*x^18 - 726*x^17 + 877*x^16 - 1452*x^15 + 156*x^14 - 272*x^13 + 3328*x^12 - 4096*x^11 + 6464*x^10 + 1024*x^9 + 768*x^8 - 12288*x^7 + 17408*x^6 - 24576*x^5 - 16384*x^3 + 49152*x^2 - 65536*x + 65536)
 
gp: K = bnfinit(y^32 - 2*y^31 + 3*y^30 - 2*y^29 - 12*y^27 + 17*y^26 - 24*y^25 + 3*y^24 + 8*y^23 + 101*y^22 - 128*y^21 + 208*y^20 - 34*y^19 + 39*y^18 - 726*y^17 + 877*y^16 - 1452*y^15 + 156*y^14 - 272*y^13 + 3328*y^12 - 4096*y^11 + 6464*y^10 + 1024*y^9 + 768*y^8 - 12288*y^7 + 17408*y^6 - 24576*y^5 - 16384*y^3 + 49152*y^2 - 65536*y + 65536, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 - 2*x^31 + 3*x^30 - 2*x^29 - 12*x^27 + 17*x^26 - 24*x^25 + 3*x^24 + 8*x^23 + 101*x^22 - 128*x^21 + 208*x^20 - 34*x^19 + 39*x^18 - 726*x^17 + 877*x^16 - 1452*x^15 + 156*x^14 - 272*x^13 + 3328*x^12 - 4096*x^11 + 6464*x^10 + 1024*x^9 + 768*x^8 - 12288*x^7 + 17408*x^6 - 24576*x^5 - 16384*x^3 + 49152*x^2 - 65536*x + 65536);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^32 - 2*x^31 + 3*x^30 - 2*x^29 - 12*x^27 + 17*x^26 - 24*x^25 + 3*x^24 + 8*x^23 + 101*x^22 - 128*x^21 + 208*x^20 - 34*x^19 + 39*x^18 - 726*x^17 + 877*x^16 - 1452*x^15 + 156*x^14 - 272*x^13 + 3328*x^12 - 4096*x^11 + 6464*x^10 + 1024*x^9 + 768*x^8 - 12288*x^7 + 17408*x^6 - 24576*x^5 - 16384*x^3 + 49152*x^2 - 65536*x + 65536)
 

\( x^{32} - 2 x^{31} + 3 x^{30} - 2 x^{29} - 12 x^{27} + 17 x^{26} - 24 x^{25} + 3 x^{24} + 8 x^{23} + \cdots + 65536 \) 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:   \(5037920877776643425304576000000000000000000000000\) \(\medspace = 2^{48}\cdot 3^{16}\cdot 5^{24}\cdot 17^{8}\) 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:  \(33.26\)
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:  $2^{3/2}3^{1/2}5^{3/4}17^{1/2}\approx 67.53946012591136$
Ramified primes:   \(2\), \(3\), \(5\), \(17\) 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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $16$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is a CM field.
Reflex fields:  unavailable$^{32768}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{15}-\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{18}-\frac{1}{4}a^{16}+\frac{1}{4}a^{12}-\frac{1}{2}a^{11}-\frac{1}{4}a^{10}-\frac{1}{2}a^{9}+\frac{1}{4}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{4}a^{4}+\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{8}a^{19}-\frac{1}{8}a^{17}-\frac{1}{2}a^{16}-\frac{1}{2}a^{15}-\frac{1}{2}a^{14}+\frac{1}{8}a^{13}+\frac{1}{4}a^{12}+\frac{3}{8}a^{11}-\frac{1}{4}a^{10}-\frac{3}{8}a^{9}+\frac{1}{4}a^{8}-\frac{1}{4}a^{6}+\frac{3}{8}a^{5}+\frac{1}{8}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{48}a^{20}+\frac{1}{16}a^{18}-\frac{1}{4}a^{17}-\frac{1}{2}a^{16}-\frac{1}{4}a^{15}+\frac{3}{16}a^{14}-\frac{1}{8}a^{13}-\frac{3}{16}a^{12}+\frac{1}{8}a^{11}+\frac{17}{48}a^{10}+\frac{3}{8}a^{9}-\frac{1}{4}a^{8}+\frac{1}{8}a^{7}+\frac{1}{16}a^{6}-\frac{1}{16}a^{4}-\frac{3}{8}a^{3}-\frac{1}{2}a^{2}+\frac{1}{3}$, $\frac{1}{96}a^{21}+\frac{1}{32}a^{19}-\frac{1}{8}a^{18}-\frac{1}{4}a^{17}-\frac{1}{8}a^{16}-\frac{13}{32}a^{15}-\frac{1}{16}a^{14}-\frac{3}{32}a^{13}+\frac{1}{16}a^{12}+\frac{17}{96}a^{11}+\frac{3}{16}a^{10}-\frac{1}{8}a^{9}-\frac{7}{16}a^{8}-\frac{15}{32}a^{7}-\frac{1}{2}a^{6}+\frac{15}{32}a^{5}+\frac{5}{16}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{3}a$, $\frac{1}{192}a^{22}-\frac{1}{192}a^{20}-\frac{1}{16}a^{19}+\frac{1}{16}a^{18}+\frac{3}{16}a^{17}-\frac{29}{64}a^{16}+\frac{7}{32}a^{15}-\frac{15}{64}a^{14}-\frac{11}{32}a^{13}-\frac{91}{192}a^{12}+\frac{15}{32}a^{11}-\frac{1}{6}a^{10}-\frac{3}{32}a^{9}-\frac{15}{64}a^{8}-\frac{3}{8}a^{7}+\frac{11}{64}a^{6}+\frac{5}{32}a^{5}-\frac{1}{16}a^{4}-\frac{3}{8}a^{3}+\frac{1}{12}a^{2}-\frac{1}{3}$, $\frac{1}{384}a^{23}-\frac{1}{384}a^{21}-\frac{1}{96}a^{20}+\frac{1}{32}a^{19}-\frac{3}{32}a^{18}+\frac{3}{128}a^{17}+\frac{23}{64}a^{16}+\frac{17}{128}a^{15}+\frac{1}{64}a^{14}+\frac{53}{384}a^{13}-\frac{13}{64}a^{12}+\frac{1}{24}a^{11}-\frac{85}{192}a^{10}+\frac{33}{128}a^{9}+\frac{5}{16}a^{8}+\frac{27}{128}a^{7}-\frac{23}{64}a^{6}-\frac{1}{32}a^{5}-\frac{1}{2}a^{4}+\frac{1}{6}a^{3}-\frac{1}{4}a^{2}-\frac{1}{6}a+\frac{1}{3}$, $\frac{1}{768}a^{24}-\frac{1}{768}a^{22}-\frac{1}{192}a^{21}-\frac{1}{192}a^{20}-\frac{3}{64}a^{19}-\frac{13}{256}a^{18}-\frac{9}{128}a^{17}-\frac{111}{256}a^{16}-\frac{31}{128}a^{15}-\frac{91}{768}a^{14}-\frac{61}{128}a^{13}-\frac{7}{24}a^{12}-\frac{133}{384}a^{11}+\frac{211}{768}a^{10}+\frac{9}{32}a^{9}-\frac{37}{256}a^{8}+\frac{25}{128}a^{7}-\frac{5}{64}a^{6}+\frac{1}{4}a^{5}+\frac{7}{48}a^{4}-\frac{1}{4}a^{3}+\frac{5}{12}a^{2}+\frac{1}{6}a-\frac{1}{3}$, $\frac{1}{1536}a^{25}-\frac{1}{1536}a^{23}-\frac{1}{384}a^{22}-\frac{1}{384}a^{21}-\frac{1}{384}a^{20}-\frac{13}{512}a^{19}+\frac{7}{256}a^{18}+\frac{17}{512}a^{17}+\frac{97}{256}a^{16}-\frac{475}{1536}a^{15}-\frac{13}{256}a^{14}+\frac{11}{48}a^{13}-\frac{277}{768}a^{12}+\frac{403}{1536}a^{11}+\frac{95}{192}a^{10}+\frac{155}{512}a^{9}+\frac{89}{256}a^{8}-\frac{53}{128}a^{7}-\frac{5}{16}a^{6}-\frac{41}{96}a^{5}+\frac{5}{16}a^{4}+\frac{1}{3}a^{3}-\frac{5}{12}a^{2}-\frac{1}{6}a+\frac{1}{3}$, $\frac{1}{1373184}a^{26}+\frac{1}{171648}a^{25}+\frac{823}{1373184}a^{24}+\frac{13}{114432}a^{23}+\frac{169}{343296}a^{22}-\frac{1523}{343296}a^{21}+\frac{8953}{1373184}a^{20}-\frac{9469}{228864}a^{19}-\frac{2807}{457728}a^{18}+\frac{10345}{228864}a^{17}+\frac{646541}{1373184}a^{16}+\frac{220865}{686592}a^{15}+\frac{14959}{171648}a^{14}-\frac{39011}{228864}a^{13}+\frac{214163}{1373184}a^{12}-\frac{4669}{21456}a^{11}+\frac{556457}{1373184}a^{10}+\frac{80441}{228864}a^{9}+\frac{937}{114432}a^{8}+\frac{11353}{57216}a^{7}+\frac{413}{2682}a^{6}+\frac{16675}{42912}a^{5}-\frac{127}{10728}a^{4}-\frac{925}{3576}a^{3}-\frac{433}{2682}a^{2}-\frac{85}{2682}a+\frac{157}{1341}$, $\frac{1}{2746368}a^{27}+\frac{253}{915456}a^{25}+\frac{181}{686592}a^{24}-\frac{143}{686592}a^{23}-\frac{1087}{686592}a^{22}+\frac{473}{2746368}a^{21}+\frac{1}{9216}a^{20}-\frac{22951}{915456}a^{19}+\frac{10767}{152576}a^{18}+\frac{278717}{2746368}a^{17}+\frac{50685}{152576}a^{16}-\frac{13447}{38144}a^{15}+\frac{301855}{1373184}a^{14}-\frac{874237}{2746368}a^{13}+\frac{115025}{343296}a^{12}-\frac{1244087}{2746368}a^{11}+\frac{85999}{1373184}a^{10}+\frac{72533}{228864}a^{9}-\frac{743}{38144}a^{8}-\frac{30667}{85824}a^{7}-\frac{521}{2384}a^{6}-\frac{1333}{3576}a^{5}-\frac{1103}{10728}a^{4}-\frac{397}{1341}a^{3}+\frac{1591}{5364}a^{2}+\frac{472}{1341}a-\frac{628}{1341}$, $\frac{1}{38449152}a^{28}+\frac{1}{19224576}a^{27}-\frac{5}{38449152}a^{26}-\frac{1837}{6408192}a^{25}+\frac{1973}{4806144}a^{24}-\frac{1643}{1373184}a^{23}-\frac{11609}{5492736}a^{22}-\frac{16127}{3204096}a^{21}-\frac{90431}{12816384}a^{20}+\frac{111989}{3204096}a^{19}-\frac{677677}{5492736}a^{18}-\frac{1827365}{9612288}a^{17}+\frac{978989}{2403072}a^{16}+\frac{1469}{43008}a^{15}-\frac{3040801}{38449152}a^{14}-\frac{1209101}{19224576}a^{13}+\frac{7150469}{38449152}a^{12}-\frac{44875}{400512}a^{11}+\frac{143827}{457728}a^{10}-\frac{583157}{1602048}a^{9}-\frac{388285}{1201536}a^{8}+\frac{42985}{300384}a^{7}+\frac{20471}{42912}a^{6}+\frac{2737}{7152}a^{5}+\frac{4721}{18774}a^{4}+\frac{937}{37548}a^{3}+\frac{3287}{18774}a^{2}-\frac{843}{2086}a+\frac{373}{1043}$, $\frac{1}{2383847424}a^{29}+\frac{13}{1191923712}a^{28}-\frac{79}{794615808}a^{27}+\frac{25}{132435968}a^{26}+\frac{18209}{74495232}a^{25}-\frac{37327}{66217984}a^{24}+\frac{48191}{37838848}a^{23}+\frac{1527193}{595961856}a^{22}+\frac{1234429}{340549632}a^{21}+\frac{1453511}{595961856}a^{20}+\frac{110473373}{2383847424}a^{19}+\frac{32293267}{595961856}a^{18}+\frac{6813511}{33108992}a^{17}+\frac{58222167}{132435968}a^{16}-\frac{1074248401}{2383847424}a^{15}+\frac{26402679}{132435968}a^{14}+\frac{338108095}{794615808}a^{13}+\frac{13246097}{148990464}a^{12}+\frac{192172733}{595961856}a^{11}-\frac{10804051}{74495232}a^{10}-\frac{34551901}{74495232}a^{9}+\frac{683359}{2660544}a^{8}+\frac{1796261}{6207936}a^{7}-\frac{5493}{73904}a^{6}+\frac{202639}{9311904}a^{5}-\frac{10553}{64666}a^{4}+\frac{31847}{96999}a^{3}+\frac{132326}{290997}a^{2}+\frac{214321}{581994}a-\frac{98162}{290997}$, $\frac{1}{4767694848}a^{30}-\frac{5}{529743872}a^{28}-\frac{83}{1191923712}a^{27}+\frac{89}{595961856}a^{26}+\frac{17813}{1191923712}a^{25}+\frac{1764617}{4767694848}a^{24}+\frac{1286669}{2383847424}a^{23}+\frac{619407}{529743872}a^{22}-\frac{1662463}{794615808}a^{21}-\frac{31592527}{4767694848}a^{20}+\frac{4017521}{264871936}a^{19}-\frac{14768321}{397307904}a^{18}-\frac{235972133}{2383847424}a^{17}-\frac{2139106717}{4767694848}a^{16}-\frac{57621611}{297980928}a^{15}+\frac{1427736349}{4767694848}a^{14}-\frac{650459153}{2383847424}a^{13}+\frac{89377301}{198653952}a^{12}+\frac{34284473}{99326976}a^{11}+\frac{138967555}{297980928}a^{10}-\frac{2848789}{16554496}a^{9}+\frac{698255}{12415872}a^{8}+\frac{2019749}{9311904}a^{7}+\frac{3603437}{9311904}a^{6}+\frac{534445}{9311904}a^{5}-\frac{1309339}{4655952}a^{4}-\frac{712855}{2327976}a^{3}-\frac{463}{193998}a^{2}+\frac{44092}{96999}a+\frac{46559}{96999}$, $\frac{1}{9535389696}a^{31}-\frac{1}{9535389696}a^{29}+\frac{17}{2383847424}a^{28}-\frac{197}{2383847424}a^{27}-\frac{611}{2383847424}a^{26}-\frac{724279}{9535389696}a^{25}-\frac{698153}{1589231616}a^{24}-\frac{978319}{3178463232}a^{23}-\frac{1402873}{4767694848}a^{22}+\frac{2443901}{1059487744}a^{21}-\frac{1527295}{227033088}a^{20}-\frac{12304021}{297980928}a^{19}-\frac{468423601}{4767694848}a^{18}-\frac{2141092141}{9535389696}a^{17}+\frac{18693379}{1191923712}a^{16}+\frac{283693441}{9535389696}a^{15}-\frac{254467377}{529743872}a^{14}+\frac{1418833}{113516544}a^{13}+\frac{343342645}{1191923712}a^{12}-\frac{32342927}{66217984}a^{11}+\frac{5905783}{33108992}a^{10}-\frac{10560919}{148990464}a^{9}-\frac{32321279}{74495232}a^{8}-\frac{3720133}{18623808}a^{7}+\frac{8143193}{18623808}a^{6}-\frac{2264785}{9311904}a^{5}-\frac{471277}{1551984}a^{4}-\frac{14351}{96999}a^{3}-\frac{363739}{1163988}a^{2}+\frac{195613}{581994}a-\frac{47933}{96999}$ 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

$C_{4}\times C_{4}$, which has order $16$ (assuming GRH)

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:   \( \frac{47}{4571136} a^{31} + \frac{5633}{340549632} a^{30} - \frac{4229}{529743872} a^{29} + \frac{43541}{2383847424} a^{28} + \frac{941}{397307904} a^{27} - \frac{38225}{198653952} a^{26} - \frac{1284011}{4767694848} a^{25} - \frac{11399}{297980928} a^{24} - \frac{158635}{681099264} a^{23} - \frac{904063}{1191923712} a^{22} + \frac{6664165}{4767694848} a^{21} + \frac{1070059}{397307904} a^{20} + \frac{3985}{4272128} a^{19} + \frac{6172933}{2383847424} a^{18} + \frac{4386553}{1589231616} a^{17} - \frac{5523821}{794615808} a^{16} - \frac{11069533}{681099264} a^{15} - \frac{83371}{10642176} a^{14} - \frac{2893957}{148990464} a^{13} - \frac{25379927}{1191923712} a^{12} + \frac{8495279}{297980928} a^{11} + \frac{583663}{8277248} a^{10} + \frac{251537}{8277248} a^{9} + \frac{1776961}{18623808} a^{8} + \frac{906827}{6207936} a^{7} - \frac{4195}{73904} a^{6} - \frac{288347}{1163988} a^{5} - \frac{43541}{1163988} a^{4} - \frac{293827}{1163988} a^{3} - \frac{441451}{1163988} a^{2} - \frac{1051}{9387} a + \frac{163204}{290997} \)  (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:   $\frac{2995}{170274816}a^{31}+\frac{3953}{56758272}a^{30}-\frac{3953}{170274816}a^{29}+\frac{3953}{170274816}a^{28}+\frac{3953}{85137408}a^{27}-\frac{223}{2364928}a^{26}-\frac{146261}{170274816}a^{25}-\frac{27671}{170274816}a^{24}-\frac{27671}{170274816}a^{23}-\frac{43483}{56758272}a^{22}-\frac{2575}{56758272}a^{21}+\frac{359723}{56758272}a^{20}+\frac{98825}{42568704}a^{19}+\frac{264851}{85137408}a^{18}+\frac{976391}{170274816}a^{17}+\frac{131973}{18919424}a^{16}-\frac{6257599}{170274816}a^{15}-\frac{1656307}{170274816}a^{14}-\frac{992203}{42568704}a^{13}-\frac{51389}{1182464}a^{12}-\frac{261171}{4729856}a^{11}+\frac{573185}{3547392}a^{10}+\frac{51389}{1330272}a^{9}+\frac{146261}{2660544}a^{8}+\frac{75107}{332568}a^{7}+\frac{73495}{295616}a^{6}-\frac{43483}{83142}a^{5}-\frac{3953}{41571}a^{4}-\frac{3953}{41571}a^{3}-\frac{3953}{13857}a^{2}-\frac{58283}{41571}a+\frac{29669}{13857}$, $\frac{27379}{340549632}a^{31}-\frac{1330361}{4767694848}a^{30}+\frac{3709}{16554496}a^{29}-\frac{12443}{227033088}a^{28}-\frac{9209}{794615808}a^{27}-\frac{1139609}{1191923712}a^{26}+\frac{6668527}{2383847424}a^{25}-\frac{3597077}{4767694848}a^{24}-\frac{1052059}{794615808}a^{23}+\frac{1521181}{4767694848}a^{22}+\frac{20808449}{2383847424}a^{21}-\frac{104713589}{4767694848}a^{20}+\frac{788495}{198653952}a^{19}+\frac{220603}{66217984}a^{18}+\frac{134381}{16554496}a^{17}-\frac{302983039}{4767694848}a^{16}+\frac{379315}{2660544}a^{15}-\frac{141166021}{4767694848}a^{14}-\frac{3960169}{397307904}a^{13}-\frac{4980001}{297980928}a^{12}+\frac{95393351}{297980928}a^{11}-\frac{6396827}{9311904}a^{10}+\frac{511547}{12415872}a^{9}+\frac{5236289}{24831744}a^{8}-\frac{20431}{775992}a^{7}-\frac{1633085}{1163988}a^{6}+\frac{11949431}{4655952}a^{5}+\frac{95251}{4655952}a^{4}-\frac{75977}{387996}a^{3}-\frac{261275}{166284}a^{2}+\frac{580592}{96999}a-\frac{2243380}{290997}$, $\frac{26249}{4767694848}a^{31}-\frac{18895}{153796608}a^{30}+\frac{1248517}{4767694848}a^{29}+\frac{480749}{4767694848}a^{28}+\frac{177749}{2383847424}a^{27}+\frac{49501}{595961856}a^{26}+\frac{1181027}{681099264}a^{25}-\frac{1103759}{529743872}a^{24}-\frac{13430177}{4767694848}a^{23}-\frac{6366721}{4767694848}a^{22}-\frac{6591799}{4767694848}a^{21}-\frac{80680687}{4767694848}a^{20}+\frac{13637087}{1191923712}a^{19}+\frac{41147033}{2383847424}a^{18}+\frac{12893387}{681099264}a^{17}+\frac{7917151}{681099264}a^{16}+\frac{585773707}{4767694848}a^{15}-\frac{29102499}{529743872}a^{14}-\frac{86784295}{1191923712}a^{13}-\frac{782729}{9612288}a^{12}-\frac{3761251}{37247616}a^{11}-\frac{50205413}{74495232}a^{10}+\frac{12234679}{74495232}a^{9}+\frac{852959}{2660544}a^{8}+\frac{2567233}{18623808}a^{7}+\frac{929255}{4655952}a^{6}+\frac{13599659}{4655952}a^{5}-\frac{5435}{64666}a^{4}-\frac{177749}{290997}a^{3}-\frac{203998}{290997}a^{2}+\frac{62035}{41571}a-\frac{361283}{41571}$, $\frac{8321}{397307904}a^{31}-\frac{8623}{85137408}a^{30}-\frac{8321}{397307904}a^{29}-\frac{8321}{595961856}a^{28}-\frac{8321}{297980928}a^{27}-\frac{91531}{297980928}a^{26}+\frac{326065}{397307904}a^{25}+\frac{158099}{297980928}a^{24}+\frac{307877}{1191923712}a^{23}+\frac{108173}{297980928}a^{22}+\frac{1206545}{397307904}a^{21}-\frac{358817}{99326976}a^{20}-\frac{108173}{33108992}a^{19}-\frac{2088571}{595961856}a^{18}-\frac{3486499}{1191923712}a^{17}-\frac{13172143}{595961856}a^{16}+\frac{10078577}{397307904}a^{15}+\frac{2055287}{148990464}a^{14}+\frac{557507}{37247616}a^{13}+\frac{208025}{9311904}a^{12}+\frac{108173}{886848}a^{11}-\frac{733981}{8277248}a^{10}-\frac{91531}{1551984}a^{9}-\frac{8321}{332568}a^{8}-\frac{8321}{166284}a^{7}-\frac{307877}{581994}a^{6}+\frac{495449}{3103968}a^{5}+\frac{33284}{290997}a^{4}+\frac{33284}{290997}a^{3}-\frac{66568}{290997}a^{2}+\frac{133136}{96999}a-\frac{378233}{290997}$, $\frac{130561}{1362198528}a^{31}-\frac{16339}{681099264}a^{30}+\frac{87091}{3178463232}a^{29}+\frac{326327}{4767694848}a^{28}-\frac{19613}{264871936}a^{27}-\frac{960095}{794615808}a^{26}-\frac{3064937}{9535389696}a^{25}-\frac{546677}{2383847424}a^{24}-\frac{1501225}{1362198528}a^{23}+\frac{519947}{2383847424}a^{22}+\frac{84923815}{9535389696}a^{21}+\frac{3283193}{794615808}a^{20}+\frac{114575}{25632768}a^{19}+\frac{40935661}{4767694848}a^{18}+\frac{3204769}{1059487744}a^{17}-\frac{27113859}{529743872}a^{16}-\frac{26280031}{1362198528}a^{15}-\frac{2879983}{85137408}a^{14}-\frac{19366609}{297980928}a^{13}-\frac{36632305}{1191923712}a^{12}+\frac{133625333}{595961856}a^{11}+\frac{3923243}{49663488}a^{10}+\frac{4198177}{49663488}a^{9}+\frac{12498007}{37247616}a^{8}+\frac{1449599}{4138624}a^{7}-\frac{315535}{443424}a^{6}-\frac{521489}{2327976}a^{5}-\frac{326327}{2327976}a^{4}-\frac{1109089}{2327976}a^{3}-\frac{2194505}{1163988}a^{2}+\frac{27383}{18774}a+\frac{65054}{290997}$, $\frac{1363147}{9535389696}a^{31}-\frac{133993}{595961856}a^{30}+\frac{2412433}{9535389696}a^{29}+\frac{354143}{2383847424}a^{28}-\frac{42185}{1191923712}a^{27}-\frac{4122709}{2383847424}a^{26}+\frac{18173123}{9535389696}a^{25}-\frac{6617881}{4767694848}a^{24}-\frac{33590611}{9535389696}a^{23}-\frac{859369}{681099264}a^{22}+\frac{127205563}{9535389696}a^{21}-\frac{23251183}{1589231616}a^{20}+\frac{3536561}{340549632}a^{19}+\frac{96162497}{4767694848}a^{18}+\frac{235519201}{9535389696}a^{17}-\frac{6044197}{74495232}a^{16}+\frac{1010837407}{9535389696}a^{15}-\frac{316023707}{4767694848}a^{14}-\frac{4430413}{38449152}a^{13}-\frac{8081765}{74495232}a^{12}+\frac{208386641}{595961856}a^{11}-\frac{55750157}{99326976}a^{10}+\frac{22340389}{148990464}a^{9}+\frac{23524001}{37247616}a^{8}+\frac{18054899}{37247616}a^{7}-\frac{1967731}{1330272}a^{6}+\frac{2583793}{1163988}a^{5}-\frac{70631}{2327976}a^{4}-\frac{2141269}{2327976}a^{3}-\frac{4097845}{1163988}a^{2}+\frac{3159743}{581994}a-\frac{1843763}{290997}$, $\frac{396563}{9535389696}a^{31}-\frac{16331}{113516544}a^{30}+\frac{286085}{9535389696}a^{29}-\frac{167}{4806144}a^{28}-\frac{13233}{264871936}a^{27}-\frac{1352011}{2383847424}a^{26}+\frac{1370435}{1059487744}a^{25}+\frac{233911}{681099264}a^{24}+\frac{56195}{3178463232}a^{23}+\frac{3425375}{4767694848}a^{22}+\frac{55371727}{9535389696}a^{21}-\frac{13747001}{1589231616}a^{20}-\frac{181775}{85137408}a^{19}-\frac{2947673}{681099264}a^{18}-\frac{9621445}{3178463232}a^{17}-\frac{103490591}{2383847424}a^{16}+\frac{151914649}{3178463232}a^{15}+\frac{61712411}{4767694848}a^{14}+\frac{828245}{37838848}a^{13}+\frac{20844791}{595961856}a^{12}+\frac{75202255}{297980928}a^{11}-\frac{2393623}{12415872}a^{10}-\frac{344491}{21284352}a^{9}+\frac{674435}{74495232}a^{8}-\frac{1550665}{12415872}a^{7}-\frac{309521}{290997}a^{6}+\frac{1356823}{3103968}a^{5}-\frac{14123}{290997}a^{4}+\frac{6569}{55428}a^{3}-\frac{722947}{1163988}a^{2}+\frac{669175}{193998}a-\frac{135214}{96999}$, $\frac{266345}{4767694848}a^{31}-\frac{449}{31997952}a^{30}+\frac{224909}{4767694848}a^{29}+\frac{753005}{4767694848}a^{28}+\frac{9323}{76898304}a^{27}-\frac{629785}{1191923712}a^{26}-\frac{49253}{681099264}a^{25}-\frac{773147}{4767694848}a^{24}-\frac{878537}{529743872}a^{23}-\frac{8207233}{4767694848}a^{22}+\frac{15219961}{4767694848}a^{21}+\frac{603217}{4767694848}a^{20}+\frac{4117291}{1191923712}a^{19}+\frac{24370079}{2383847424}a^{18}+\frac{9936467}{681099264}a^{17}-\frac{9328661}{681099264}a^{16}+\frac{33065387}{4767694848}a^{15}-\frac{81577339}{4767694848}a^{14}-\frac{7863937}{132435968}a^{13}-\frac{101399093}{1191923712}a^{12}+\frac{6920029}{148990464}a^{11}-\frac{4231327}{74495232}a^{10}-\frac{5490101}{148990464}a^{9}+\frac{592637}{2660544}a^{8}+\frac{7034887}{18623808}a^{7}-\frac{871699}{4655952}a^{6}+\frac{1235893}{4655952}a^{5}+\frac{291457}{4655952}a^{4}-\frac{7039}{64666}a^{3}-\frac{336454}{290997}a^{2}+\frac{1627}{4619}a-\frac{14465}{13857}$, $\frac{22595}{3178463232}a^{31}-\frac{4495}{51265536}a^{30}+\frac{592561}{3178463232}a^{29}+\frac{31385}{529743872}a^{28}+\frac{58375}{1191923712}a^{27}+\frac{30145}{2383847424}a^{26}+\frac{1632335}{1362198528}a^{25}-\frac{466691}{264871936}a^{24}-\frac{17542645}{9535389696}a^{23}-\frac{2024305}{2383847424}a^{22}-\frac{5295715}{9535389696}a^{21}-\frac{9187415}{794615808}a^{20}+\frac{1242761}{99326976}a^{19}+\frac{17828885}{1589231616}a^{18}+\frac{17122945}{1362198528}a^{17}+\frac{3473185}{681099264}a^{16}+\frac{798708445}{9535389696}a^{15}-\frac{2126629}{33108992}a^{14}-\frac{112705385}{2383847424}a^{13}-\frac{1046495}{19224576}a^{12}-\frac{2037545}{37247616}a^{11}-\frac{1891215}{4138624}a^{10}+\frac{2979679}{16554496}a^{9}+\frac{5935}{27714}a^{8}+\frac{3472355}{37247616}a^{7}+\frac{102635}{1163988}a^{6}+\frac{18431345}{9311904}a^{5}-\frac{113737}{517328}a^{4}-\frac{116750}{290997}a^{3}-\frac{301285}{581994}a^{2}+\frac{95765}{83142}a-\frac{24390}{4619}$, $\frac{319601}{4767694848}a^{31}+\frac{32783}{595961856}a^{30}-\frac{203641}{1589231616}a^{29}-\frac{64853}{1191923712}a^{28}-\frac{48383}{340549632}a^{27}-\frac{959809}{1191923712}a^{26}-\frac{5980999}{4767694848}a^{25}+\frac{2335141}{2383847424}a^{24}+\frac{7912585}{4767694848}a^{23}-\frac{135281}{340549632}a^{22}+\frac{35022175}{4767694848}a^{21}+\frac{29346635}{2383847424}a^{20}-\frac{1966879}{794615808}a^{19}-\frac{16543859}{2383847424}a^{18}-\frac{3228965}{4767694848}a^{17}-\frac{52905107}{1191923712}a^{16}-\frac{429059741}{4767694848}a^{15}+\frac{9426083}{2383847424}a^{14}+\frac{27623377}{2383847424}a^{13}-\frac{5064029}{1191923712}a^{12}+\frac{61570367}{297980928}a^{11}+\frac{143326415}{297980928}a^{10}+\frac{25709}{886848}a^{9}-\frac{2429677}{37247616}a^{8}+\frac{15530005}{37247616}a^{7}-\frac{2029639}{2660544}a^{6}-\frac{9168451}{4655952}a^{5}-\frac{829907}{2327976}a^{4}+\frac{23063}{332568}a^{3}-\frac{309028}{290997}a^{2}+\frac{12749}{32333}a+\frac{1685126}{290997}$, $\frac{35}{305152}a^{31}-\frac{29389}{132435968}a^{30}+\frac{780659}{2383847424}a^{29}+\frac{46289}{1191923712}a^{28}+\frac{72193}{2383847424}a^{27}-\frac{600785}{595961856}a^{26}+\frac{71803}{37247616}a^{25}-\frac{2296163}{1191923712}a^{24}-\frac{5907017}{2383847424}a^{23}-\frac{431099}{297980928}a^{22}+\frac{2235539}{340549632}a^{21}-\frac{292081}{16554496}a^{20}+\frac{28444111}{2383847424}a^{19}+\frac{15149875}{1191923712}a^{18}+\frac{90961}{4655952}a^{17}-\frac{21434167}{595961856}a^{16}+\frac{297704345}{2383847424}a^{15}-\frac{72981625}{1191923712}a^{14}-\frac{158355829}{2383847424}a^{13}-\frac{115205791}{1191923712}a^{12}+\frac{68773465}{595961856}a^{11}-\frac{21148105}{33108992}a^{10}+\frac{2106377}{21284352}a^{9}+\frac{31617941}{74495232}a^{8}+\frac{11953639}{37247616}a^{7}-\frac{9153097}{18623808}a^{6}+\frac{802925}{290997}a^{5}-\frac{347581}{2327976}a^{4}-\frac{279551}{2327976}a^{3}-\frac{669092}{290997}a^{2}+\frac{1024243}{290997}a-\frac{227734}{32333}$, $\frac{1308697}{9535389696}a^{31}+\frac{209863}{529743872}a^{30}-\frac{6719165}{9535389696}a^{29}+\frac{61823}{529743872}a^{28}+\frac{3881}{56758272}a^{27}-\frac{3318863}{2383847424}a^{26}-\frac{60177431}{9535389696}a^{25}+\frac{5073211}{1191923712}a^{24}+\frac{36128867}{9535389696}a^{23}-\frac{1587013}{1191923712}a^{22}+\frac{23391799}{3178463232}a^{21}+\frac{11292391}{198653952}a^{20}-\frac{20175143}{1191923712}a^{19}-\frac{8594865}{529743872}a^{18}-\frac{3057801}{1059487744}a^{17}-\frac{154012403}{4767694848}a^{16}-\frac{512285077}{1362198528}a^{15}+\frac{206933941}{2383847424}a^{14}+\frac{6298909}{2383847424}a^{13}-\frac{40892977}{595961856}a^{12}+\frac{4492725}{33108992}a^{11}+\frac{26872867}{14189568}a^{10}-\frac{41522375}{148990464}a^{9}-\frac{750493}{4138624}a^{8}+\frac{881167}{886848}a^{7}+\frac{1557959}{2660544}a^{6}-\frac{71535113}{9311904}a^{5}-\frac{557717}{4655952}a^{4}+\frac{891671}{581994}a^{3}-\frac{392731}{290997}a^{2}-\frac{2593264}{290997}a+\frac{1982042}{96999}$, $\frac{1000729}{3178463232}a^{31}-\frac{101033}{681099264}a^{30}+\frac{6264289}{9535389696}a^{29}+\frac{1237037}{4767694848}a^{28}+\frac{44521}{99326976}a^{27}-\frac{8014075}{2383847424}a^{26}+\frac{221293}{1362198528}a^{25}-\frac{4203869}{595961856}a^{24}-\frac{72422927}{9535389696}a^{23}-\frac{10595779}{1191923712}a^{22}+\frac{192922679}{9535389696}a^{21}-\frac{610253}{85137408}a^{20}+\frac{34438829}{595961856}a^{19}+\frac{296066713}{4767694848}a^{18}+\frac{45770521}{454066176}a^{17}-\frac{463830517}{4767694848}a^{16}+\frac{1119227423}{9535389696}a^{15}-\frac{757143757}{2383847424}a^{14}-\frac{855951631}{2383847424}a^{13}-\frac{736354313}{1191923712}a^{12}+\frac{4675045}{19224576}a^{11}-\frac{237491501}{297980928}a^{10}+\frac{76599203}{74495232}a^{9}+\frac{60476587}{37247616}a^{8}+\frac{16594369}{6207936}a^{7}-\frac{1693735}{9311904}a^{6}+\frac{5364887}{1163988}a^{5}-\frac{2474695}{1163988}a^{4}-\frac{5835239}{2327976}a^{3}-\frac{5265881}{581994}a^{2}+\frac{692674}{290997}a-\frac{1593842}{96999}$, $\frac{114617}{340549632}a^{31}+\frac{423337}{2383847424}a^{30}+\frac{6341}{99326976}a^{29}+\frac{25663}{76898304}a^{28}+\frac{150965}{264871936}a^{27}-\frac{4057637}{1191923712}a^{26}-\frac{1395187}{340549632}a^{25}-\frac{1492663}{794615808}a^{24}-\frac{4884689}{1191923712}a^{23}-\frac{3173561}{340549632}a^{22}+\frac{8167361}{397307904}a^{21}+\frac{74011031}{2383847424}a^{20}+\frac{19738903}{794615808}a^{19}+\frac{43272835}{1191923712}a^{18}+\frac{62224369}{794615808}a^{17}-\frac{1640071}{15998976}a^{16}-\frac{24439481}{148990464}a^{15}-\frac{38069235}{264871936}a^{14}-\frac{620223131}{2383847424}a^{13}-\frac{303578845}{595961856}a^{12}+\frac{10068241}{28379136}a^{11}+\frac{211870511}{297980928}a^{10}+\frac{7250941}{16554496}a^{9}+\frac{41541715}{37247616}a^{8}+\frac{33904477}{12415872}a^{7}-\frac{3112087}{9311904}a^{6}-\frac{18028069}{9311904}a^{5}-\frac{1620013}{1551984}a^{4}-\frac{1327799}{1163988}a^{3}-\frac{4652411}{581994}a^{2}-\frac{1250617}{581994}a+\frac{211751}{96999}$, $\frac{2163803}{9535389696}a^{31}-\frac{1195}{17088512}a^{30}-\frac{150497}{1362198528}a^{29}+\frac{277327}{4767694848}a^{28}-\frac{22073}{132435968}a^{27}-\frac{6382513}{2383847424}a^{26}-\frac{2088181}{9535389696}a^{25}+\frac{3143813}{2383847424}a^{24}-\frac{2009221}{3178463232}a^{23}+\frac{303497}{794615808}a^{22}+\frac{23253375}{1059487744}a^{21}+\frac{13393747}{2383847424}a^{20}-\frac{80341}{148990464}a^{19}+\frac{10404965}{4767694848}a^{18}+\frac{1203821}{1059487744}a^{17}-\frac{94903633}{681099264}a^{16}-\frac{377766569}{9535389696}a^{15}-\frac{23558035}{1191923712}a^{14}-\frac{40247761}{794615808}a^{13}-\frac{1008947}{99326976}a^{12}+\frac{125210321}{198653952}a^{11}+\frac{18779101}{74495232}a^{10}+\frac{6987067}{148990464}a^{9}+\frac{24022417}{74495232}a^{8}+\frac{892201}{2069312}a^{7}-\frac{42631277}{18623808}a^{6}-\frac{4931783}{4655952}a^{5}-\frac{2779823}{4655952}a^{4}+\frac{1501}{18476}a^{3}-\frac{85196}{32333}a^{2}+\frac{3154757}{581994}a+\frac{124967}{41571}$ Copy content Toggle raw display (assuming GRH)
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:  \( 190465720691.2432 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 190465720691.2432 \cdot 16}{30\cdot\sqrt{5037920877776643425304576000000000000000000000000}}\cr\approx \mathstrut & 0.267034746152232 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^32 - 2*x^31 + 3*x^30 - 2*x^29 - 12*x^27 + 17*x^26 - 24*x^25 + 3*x^24 + 8*x^23 + 101*x^22 - 128*x^21 + 208*x^20 - 34*x^19 + 39*x^18 - 726*x^17 + 877*x^16 - 1452*x^15 + 156*x^14 - 272*x^13 + 3328*x^12 - 4096*x^11 + 6464*x^10 + 1024*x^9 + 768*x^8 - 12288*x^7 + 17408*x^6 - 24576*x^5 - 16384*x^3 + 49152*x^2 - 65536*x + 65536)
 
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 + 3*x^30 - 2*x^29 - 12*x^27 + 17*x^26 - 24*x^25 + 3*x^24 + 8*x^23 + 101*x^22 - 128*x^21 + 208*x^20 - 34*x^19 + 39*x^18 - 726*x^17 + 877*x^16 - 1452*x^15 + 156*x^14 - 272*x^13 + 3328*x^12 - 4096*x^11 + 6464*x^10 + 1024*x^9 + 768*x^8 - 12288*x^7 + 17408*x^6 - 24576*x^5 - 16384*x^3 + 49152*x^2 - 65536*x + 65536, 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 + 3*x^30 - 2*x^29 - 12*x^27 + 17*x^26 - 24*x^25 + 3*x^24 + 8*x^23 + 101*x^22 - 128*x^21 + 208*x^20 - 34*x^19 + 39*x^18 - 726*x^17 + 877*x^16 - 1452*x^15 + 156*x^14 - 272*x^13 + 3328*x^12 - 4096*x^11 + 6464*x^10 + 1024*x^9 + 768*x^8 - 12288*x^7 + 17408*x^6 - 24576*x^5 - 16384*x^3 + 49152*x^2 - 65536*x + 65536);
 
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 + 3*x^30 - 2*x^29 - 12*x^27 + 17*x^26 - 24*x^25 + 3*x^24 + 8*x^23 + 101*x^22 - 128*x^21 + 208*x^20 - 34*x^19 + 39*x^18 - 726*x^17 + 877*x^16 - 1452*x^15 + 156*x^14 - 272*x^13 + 3328*x^12 - 4096*x^11 + 6464*x^10 + 1024*x^9 + 768*x^8 - 12288*x^7 + 17408*x^6 - 24576*x^5 - 16384*x^3 + 49152*x^2 - 65536*x + 65536);
 
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

$C_4^2:C_2^2$ (as 32T204):

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)
 
A solvable group of order 64
The 40 conjugacy class representatives for $C_4^2:C_2^2$
Character table for $C_4^2:C_2^2$

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{-30}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-15}) \), \(\Q(\zeta_{15})^+\), 4.0.8000.2, \(\Q(\zeta_{5})\), 4.4.72000.1, \(\Q(\sqrt{2}, \sqrt{-15})\), 4.0.27200.2, 4.4.9792.1, \(\Q(\sqrt{-3}, \sqrt{10})\), \(\Q(\sqrt{5}, \sqrt{-6})\), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\sqrt{2}, \sqrt{5})\), 4.4.244800.1, 4.0.1088.2, \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\sqrt{-6}, \sqrt{10})\), 8.0.59927040000.35, 8.0.207360000.1, 8.0.59927040000.32, 8.0.59927040000.12, 8.0.739840000.6, 8.0.95883264.1, 8.8.59927040000.2, 8.0.5184000000.1, 8.0.5184000000.5, 8.0.1498176000000.24, 8.8.18496000000.1, 8.8.5184000000.1, 8.0.64000000.2, \(\Q(\zeta_{15})\), 8.0.5184000000.3, 16.0.3591250123161600000000.1, 16.0.2244531326976000000000000.3, 16.0.26873856000000000000.2, 16.0.2244531326976000000000000.2, 16.0.342102016000000000000.1, 16.0.2244531326976000000000000.1, 16.16.2244531326976000000000000.1

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]
 

Sibling fields

Degree 32 siblings: data not computed
Minimal sibling: This field is its own minimal sibling

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 ${\href{/padicField/7.4.0.1}{4} }^{8}$ ${\href{/padicField/11.4.0.1}{4} }^{8}$ ${\href{/padicField/13.4.0.1}{4} }^{8}$ R ${\href{/padicField/19.2.0.1}{2} }^{16}$ ${\href{/padicField/23.4.0.1}{4} }^{8}$ ${\href{/padicField/29.4.0.1}{4} }^{8}$ ${\href{/padicField/31.2.0.1}{2} }^{8}{,}\,{\href{/padicField/31.1.0.1}{1} }^{16}$ ${\href{/padicField/37.4.0.1}{4} }^{8}$ ${\href{/padicField/41.2.0.1}{2} }^{16}$ ${\href{/padicField/43.4.0.1}{4} }^{8}$ ${\href{/padicField/47.4.0.1}{4} }^{8}$ ${\href{/padicField/53.4.0.1}{4} }^{8}$ ${\href{/padicField/59.2.0.1}{2} }^{16}$

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 2.8.12.1$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}$
\(3\) 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 5.16.12.1$x^{16} + 16 x^{14} + 16 x^{13} + 124 x^{12} + 192 x^{11} + 368 x^{10} - 16 x^{9} + 1462 x^{8} + 2688 x^{7} + 2544 x^{6} + 5232 x^{5} + 14108 x^{4} + 4800 x^{3} + 8592 x^{2} - 6512 x + 13041$$4$$4$$12$$C_4^2$$[\ ]_{4}^{4}$
5.16.12.1$x^{16} + 16 x^{14} + 16 x^{13} + 124 x^{12} + 192 x^{11} + 368 x^{10} - 16 x^{9} + 1462 x^{8} + 2688 x^{7} + 2544 x^{6} + 5232 x^{5} + 14108 x^{4} + 4800 x^{3} + 8592 x^{2} - 6512 x + 13041$$4$$4$$12$$C_4^2$$[\ ]_{4}^{4}$
\(17\) Copy content Toggle raw display 17.4.0.1$x^{4} + 7 x^{2} + 10 x + 3$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.0.1$x^{4} + 7 x^{2} + 10 x + 3$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.0.1$x^{4} + 7 x^{2} + 10 x + 3$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.0.1$x^{4} + 7 x^{2} + 10 x + 3$$1$$4$$0$$C_4$$[\ ]^{4}$
17.8.4.1$x^{8} + 612 x^{7} + 140536 x^{6} + 14363966 x^{5} + 553913435 x^{4} + 345855654 x^{3} + 4032327212 x^{2} + 6379401496 x + 2294776272$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
17.8.4.1$x^{8} + 612 x^{7} + 140536 x^{6} + 14363966 x^{5} + 553913435 x^{4} + 345855654 x^{3} + 4032327212 x^{2} + 6379401496 x + 2294776272$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$