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Properties

Label 32.0.794...000.1
Degree $32$
Signature $[0, 16]$
Discriminant $7.941\times 10^{50}$
Root discriminant \(38.96\)
Ramified primes $2,3,5$
Class number $100$ (GRH)
Class group [5, 20] (GRH)
Galois group $C_2\times C_4^2$ (as 32T36)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 4*x^30 + 2*x^28 + 40*x^26 - 160*x^24 + 544*x^22 - 264*x^20 - 5440*x^18 + 21744*x^16 - 10880*x^14 - 1056*x^12 + 4352*x^10 - 2560*x^8 + 1280*x^6 + 128*x^4 - 512*x^2 + 256)
 
gp: K = bnfinit(y^32 - 4*y^30 + 2*y^28 + 40*y^26 - 160*y^24 + 544*y^22 - 264*y^20 - 5440*y^18 + 21744*y^16 - 10880*y^14 - 1056*y^12 + 4352*y^10 - 2560*y^8 + 1280*y^6 + 128*y^4 - 512*y^2 + 256, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 - 4*x^30 + 2*x^28 + 40*x^26 - 160*x^24 + 544*x^22 - 264*x^20 - 5440*x^18 + 21744*x^16 - 10880*x^14 - 1056*x^12 + 4352*x^10 - 2560*x^8 + 1280*x^6 + 128*x^4 - 512*x^2 + 256);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^32 - 4*x^30 + 2*x^28 + 40*x^26 - 160*x^24 + 544*x^22 - 264*x^20 - 5440*x^18 + 21744*x^16 - 10880*x^14 - 1056*x^12 + 4352*x^10 - 2560*x^8 + 1280*x^6 + 128*x^4 - 512*x^2 + 256)
 

\( x^{32} - 4 x^{30} + 2 x^{28} + 40 x^{26} - 160 x^{24} + 544 x^{22} - 264 x^{20} - 5440 x^{18} + \cdots + 256 \) 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:   \(794071845499378503449051136000000000000000000000000\) \(\medspace = 2^{88}\cdot 3^{16}\cdot 5^{24}\) 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:  \(38.96\)
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:  $2^{11/4}3^{1/2}5^{3/4}\approx 38.96014985701187$
Ramified primes:   \(2\), \(3\), \(5\) 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:  \(240=2^{4}\cdot 3\cdot 5\)
Dirichlet character group:    $\lbrace$$\chi_{240}(1,·)$, $\chi_{240}(133,·)$, $\chi_{240}(137,·)$, $\chi_{240}(13,·)$, $\chi_{240}(17,·)$, $\chi_{240}(149,·)$, $\chi_{240}(89,·)$, $\chi_{240}(29,·)$, $\chi_{240}(161,·)$, $\chi_{240}(37,·)$, $\chi_{240}(41,·)$, $\chi_{240}(173,·)$, $\chi_{240}(157,·)$, $\chi_{240}(49,·)$, $\chi_{240}(181,·)$, $\chi_{240}(61,·)$, $\chi_{240}(53,·)$, $\chi_{240}(193,·)$, $\chi_{240}(197,·)$, $\chi_{240}(73,·)$, $\chi_{240}(77,·)$, $\chi_{240}(209,·)$, $\chi_{240}(217,·)$, $\chi_{240}(221,·)$, $\chi_{240}(101,·)$, $\chi_{240}(97,·)$, $\chi_{240}(229,·)$, $\chi_{240}(233,·)$, $\chi_{240}(109,·)$, $\chi_{240}(113,·)$, $\chi_{240}(169,·)$, $\chi_{240}(121,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{32768}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{2}a^{6}$, $\frac{1}{2}a^{7}$, $\frac{1}{4}a^{8}$, $\frac{1}{4}a^{9}$, $\frac{1}{4}a^{10}$, $\frac{1}{4}a^{11}$, $\frac{1}{8}a^{12}$, $\frac{1}{8}a^{13}$, $\frac{1}{8}a^{14}$, $\frac{1}{8}a^{15}$, $\frac{1}{16}a^{16}$, $\frac{1}{16}a^{17}$, $\frac{1}{16}a^{18}$, $\frac{1}{16}a^{19}$, $\frac{1}{1312}a^{20}-\frac{3}{41}a^{10}+\frac{1}{41}$, $\frac{1}{1312}a^{21}-\frac{3}{41}a^{11}+\frac{1}{41}a$, $\frac{1}{1312}a^{22}+\frac{17}{328}a^{12}+\frac{1}{41}a^{2}$, $\frac{1}{1312}a^{23}+\frac{17}{328}a^{13}+\frac{1}{41}a^{3}$, $\frac{1}{18368}a^{24}+\frac{1}{56}a^{18}+\frac{5}{164}a^{14}-\frac{1}{56}a^{12}+\frac{1}{7}a^{6}+\frac{3}{41}a^{4}-\frac{1}{7}$, $\frac{1}{18368}a^{25}+\frac{1}{56}a^{19}+\frac{5}{164}a^{15}-\frac{1}{56}a^{13}+\frac{1}{7}a^{7}+\frac{3}{41}a^{5}-\frac{1}{7}a$, $\frac{1}{17651648}a^{26}-\frac{25}{4412912}a^{24}+\frac{165}{1260832}a^{22}+\frac{2803}{8825824}a^{20}-\frac{165}{107632}a^{18}+\frac{825}{157604}a^{16}-\frac{2803}{1103228}a^{14}-\frac{165}{551614}a^{12}-\frac{11371}{157604}a^{10}-\frac{2803}{26908}a^{8}+\frac{165}{551614}a^{6}+\frac{2803}{78802}a^{4}-\frac{34}{275807}a^{2}-\frac{16221}{275807}$, $\frac{1}{17651648}a^{27}-\frac{25}{4412912}a^{25}+\frac{165}{1260832}a^{23}+\frac{2803}{8825824}a^{21}-\frac{165}{107632}a^{19}+\frac{825}{157604}a^{17}-\frac{2803}{1103228}a^{15}-\frac{165}{551614}a^{13}-\frac{11371}{157604}a^{11}-\frac{2803}{26908}a^{9}+\frac{165}{551614}a^{7}+\frac{2803}{78802}a^{5}-\frac{34}{275807}a^{3}-\frac{16221}{275807}a$, $\frac{1}{35303296}a^{28}-\frac{1}{17651648}a^{24}+\frac{493}{2206456}a^{22}-\frac{165}{1260832}a^{20}-\frac{2873}{1103228}a^{18}+\frac{563}{53816}a^{16}-\frac{825}{157604}a^{14}-\frac{1121}{2206456}a^{12}-\frac{19618}{275807}a^{10}+\frac{3283}{78802}a^{8}-\frac{981}{6727}a^{6}+\frac{19618}{275807}a^{4}-\frac{9570}{39401}a^{2}+\frac{34}{275807}$, $\frac{1}{35303296}a^{29}-\frac{1}{17651648}a^{25}+\frac{493}{2206456}a^{23}-\frac{165}{1260832}a^{21}-\frac{2873}{1103228}a^{19}+\frac{563}{53816}a^{17}-\frac{825}{157604}a^{15}-\frac{1121}{2206456}a^{13}-\frac{19618}{275807}a^{11}+\frac{3283}{78802}a^{9}-\frac{981}{6727}a^{7}+\frac{19618}{275807}a^{5}-\frac{9570}{39401}a^{3}+\frac{34}{275807}a$, $\frac{1}{35303296}a^{30}-\frac{114243}{275807}$, $\frac{1}{35303296}a^{31}-\frac{114243}{275807}a$ 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:  $31$

Class group and class number

$C_{5}\times C_{20}$, which has order $100$ (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:   \( -\frac{1479}{551614} a^{30} + \frac{1479}{157604} a^{28} + \frac{51}{2206456} a^{26} - \frac{1479}{13454} a^{24} + \frac{14790}{39401} a^{22} - \frac{343128}{275807} a^{20} - \frac{5916}{275807} a^{18} + \frac{9424999}{630416} a^{16} - \frac{343128}{6727} a^{14} + \frac{5916}{275807} a^{12} + \frac{686256}{39401} a^{10} - \frac{2827848}{275807} a^{8} + \frac{1656276}{275807} a^{6} - \frac{567936}{275807} a^{2} + \frac{47328}{39401} \)  (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:   $\frac{5771}{2521664}a^{30}-\frac{46367}{5043328}a^{28}+\frac{5771}{1260832}a^{26}+\frac{28855}{315208}a^{24}-\frac{28855}{78802}a^{22}+\frac{98107}{78802}a^{20}-\frac{196209}{315208}a^{18}-\frac{490535}{39401}a^{16}+\frac{7842789}{157604}a^{14}-\frac{981070}{39401}a^{12}-\frac{190443}{78802}a^{10}+\frac{231039}{157604}a^{8}-\frac{230840}{39401}a^{6}+\frac{115420}{39401}a^{4}+\frac{11542}{39401}a^{2}-\frac{6767}{39401}$, $\frac{1}{17651648}a^{30}-\frac{1}{35303296}a^{28}-\frac{5}{8825824}a^{26}+\frac{5}{2206456}a^{24}-\frac{17}{2206456}a^{22}+\frac{33}{8825824}a^{20}+\frac{85}{1103228}a^{18}-\frac{1359}{4412912}a^{16}+\frac{85}{551614}a^{14}+\frac{33}{2206456}a^{12}-\frac{17}{275807}a^{10}+\frac{10}{275807}a^{8}-\frac{5}{275807}a^{6}-\frac{1}{551614}a^{4}-\frac{470830}{275807}a^{2}+\frac{275806}{275807}$, $\frac{1479}{551614}a^{30}-\frac{1479}{157604}a^{28}-\frac{51}{2206456}a^{26}+\frac{1479}{13454}a^{24}-\frac{14790}{39401}a^{22}+\frac{343128}{275807}a^{20}+\frac{5916}{275807}a^{18}-\frac{9424999}{630416}a^{16}+\frac{343128}{6727}a^{14}-\frac{5916}{275807}a^{12}-\frac{686256}{39401}a^{10}+\frac{2827848}{275807}a^{8}-\frac{1656276}{275807}a^{6}+\frac{567936}{275807}a^{2}-\frac{86729}{39401}$, $\frac{985}{35303296}a^{30}+\frac{17}{1312}a^{20}+\frac{985}{164}a^{10}+\frac{114260}{275807}$, $\frac{145}{315208}a^{31}+\frac{40391}{17651648}a^{30}-\frac{145}{630416}a^{29}-\frac{323129}{35303296}a^{28}-\frac{725}{157604}a^{27}+\frac{40391}{8825824}a^{26}+\frac{725}{39401}a^{25}+\frac{201955}{2206456}a^{24}-\frac{5805}{630416}a^{23}-\frac{201955}{551614}a^{22}+\frac{4785}{157604}a^{21}+\frac{686647}{551614}a^{20}+\frac{24650}{39401}a^{19}-\frac{1332903}{2206456}a^{18}-\frac{197055}{78802}a^{17}-\frac{3433235}{275807}a^{16}+\frac{49300}{39401}a^{15}+\frac{54891369}{1103228}a^{14}+\frac{7840639}{315208}a^{13}-\frac{6866470}{275807}a^{12}-\frac{19720}{39401}a^{11}-\frac{1332903}{551614}a^{10}+\frac{11600}{39401}a^{9}+\frac{2746588}{275807}a^{8}-\frac{5800}{39401}a^{7}-\frac{1615640}{275807}a^{6}-\frac{580}{39401}a^{5}+\frac{807820}{275807}a^{4}+\frac{2330}{39401}a^{3}+\frac{80782}{275807}a^{2}-\frac{1160}{39401}a-\frac{323128}{275807}$, $\frac{3}{17651648}a^{31}-\frac{5771}{2521664}a^{30}-\frac{3}{35303296}a^{29}+\frac{46367}{5043328}a^{28}-\frac{15}{8825824}a^{27}-\frac{5771}{1260832}a^{26}+\frac{15}{2206456}a^{25}-\frac{28855}{315208}a^{24}-\frac{51}{2206456}a^{23}+\frac{28855}{78802}a^{22}+\frac{99}{8825824}a^{21}-\frac{98107}{78802}a^{20}+\frac{255}{1103228}a^{19}+\frac{196209}{315208}a^{18}-\frac{4077}{4412912}a^{17}+\frac{490535}{39401}a^{16}+\frac{255}{551614}a^{15}-\frac{7842789}{157604}a^{14}+\frac{99}{2206456}a^{13}+\frac{981070}{39401}a^{12}-\frac{51}{275807}a^{11}+\frac{190443}{78802}a^{10}+\frac{30}{275807}a^{9}-\frac{231039}{157604}a^{8}-\frac{15}{275807}a^{7}+\frac{230840}{39401}a^{6}-\frac{3}{551614}a^{5}-\frac{115420}{39401}a^{4}-\frac{1136683}{275807}a^{3}-\frac{11542}{39401}a^{2}-\frac{3}{275807}a+\frac{46168}{39401}$, $\frac{78397}{35303296}a^{30}-\frac{323105}{35303296}a^{28}+\frac{40391}{8825824}a^{26}+\frac{25244}{275807}a^{24}-\frac{3231041}{8825824}a^{22}+\frac{261213}{215264}a^{20}-\frac{2665807}{4412912}a^{18}-\frac{13731751}{1103228}a^{16}+\frac{109781345}{2206456}a^{14}-\frac{13732939}{551614}a^{12}-\frac{113795}{6727}a^{10}+\frac{2746553}{275807}a^{8}-\frac{3231239}{551614}a^{6}+\frac{2281497}{551614}a^{4}+\frac{80775}{275807}a^{2}-\frac{323165}{275807}$, $\frac{8119}{17651648}a^{31}-\frac{12129}{4412912}a^{30}-\frac{8119}{35303296}a^{29}+\frac{332687}{35303296}a^{28}-\frac{40595}{8825824}a^{27}-\frac{5}{8825824}a^{26}+\frac{40595}{2206456}a^{25}-\frac{1940401}{17651648}a^{24}-\frac{40601}{4412912}a^{23}+\frac{1656475}{4412912}a^{22}+\frac{267927}{8825824}a^{21}-\frac{11255911}{8825824}a^{20}+\frac{690115}{1103228}a^{19}-\frac{6761}{2206456}a^{18}-\frac{11033721}{4412912}a^{17}+\frac{65926313}{4412912}a^{16}+\frac{690115}{551614}a^{15}-\frac{56272823}{1103228}a^{14}+\frac{6860555}{275807}a^{13}+\frac{55481}{2206456}a^{12}-\frac{138023}{275807}a^{11}+\frac{803979}{275807}a^{10}+\frac{81190}{275807}a^{9}-\frac{485160}{275807}a^{8}-\frac{40595}{275807}a^{7}+\frac{284166}{275807}a^{6}-\frac{8119}{551614}a^{5}-\frac{120}{275807}a^{4}+\frac{487140}{275807}a^{3}-\frac{567864}{275807}a^{2}-\frac{8119}{275807}a+\frac{7921}{39401}$, $\frac{2871}{17651648}a^{31}-\frac{48511}{17651648}a^{30}+\frac{20703}{2206456}a^{28}+\frac{3}{4412912}a^{26}-\frac{1940201}{17651648}a^{24}+\frac{1656269}{4412912}a^{22}+\frac{99}{1312}a^{21}-\frac{1406789}{1103228}a^{20}-\frac{47491}{2206456}a^{18}+\frac{8239939}{551614}a^{16}-\frac{56264839}{1103228}a^{14}+\frac{5911}{275807}a^{12}+\frac{5741}{164}a^{11}+\frac{803897}{275807}a^{10}-\frac{11309759}{1103228}a^{8}+\frac{283931}{275807}a^{6}+\frac{1}{551614}a^{4}-\frac{97022}{275807}a^{2}+\frac{99}{275807}a+\frac{55400}{275807}$, $\frac{37813}{35303296}a^{30}-\frac{135815}{35303296}a^{28}+\frac{389}{17651648}a^{26}+\frac{57375}{1260832}a^{24}-\frac{343031}{2206456}a^{22}+\frac{4386009}{8825824}a^{20}+\frac{139}{5084}a^{18}-\frac{3407447}{551614}a^{16}+\frac{5823017}{275807}a^{14}-\frac{1631}{315208}a^{12}-\frac{17326475}{1103228}a^{10}+\frac{14052645}{1103228}a^{8}+\frac{179271}{39401}a^{6}-\frac{3215283}{551614}a^{4}+\frac{40121}{275807}a^{2}+\frac{252899}{275807}$, $\frac{97527}{17651648}a^{31}+\frac{1}{2206456}a^{30}-\frac{783579}{35303296}a^{29}-\frac{25}{17651648}a^{28}+\frac{97527}{8825824}a^{27}-\frac{13}{551614}a^{26}+\frac{487635}{2206456}a^{25}+\frac{327}{17651648}a^{24}-\frac{487635}{551614}a^{23}-\frac{9}{142352}a^{22}+\frac{1657959}{551614}a^{21}+\frac{1657}{8825824}a^{20}-\frac{6631865}{4412912}a^{19}+\frac{177}{2206456}a^{18}-\frac{8289795}{275807}a^{17}-\frac{58199}{4412912}a^{16}+\frac{132539193}{1103228}a^{15}+\frac{8493}{1103228}a^{14}-\frac{16579590}{275807}a^{13}+\frac{13}{1103228}a^{12}-\frac{3218391}{551614}a^{11}-\frac{1455}{551614}a^{10}+\frac{3904443}{1103228}a^{9}+\frac{1713}{1103228}a^{8}-\frac{3901080}{275807}a^{7}-\frac{1372355}{275807}a^{6}+\frac{1950540}{275807}a^{5}-\frac{1}{551614}a^{4}+\frac{195054}{275807}a^{3}-\frac{470746}{275807}a^{2}-\frac{780216}{275807}a-\frac{50}{275807}$, $\frac{19603}{17651648}a^{30}-\frac{68903}{17651648}a^{28}+\frac{13}{551614}a^{26}+\frac{57393}{1260832}a^{24}-\frac{686129}{4412912}a^{22}+\frac{284257}{551614}a^{20}+\frac{385}{315208}a^{18}-\frac{27260265}{4412912}a^{16}+\frac{46604113}{2206456}a^{14}-\frac{53}{5084}a^{12}-\frac{7955793}{1103228}a^{10}+\frac{802447}{1103228}a^{8}+\frac{179229}{39401}a^{6}-\frac{1607421}{551614}a^{4}+\frac{235147}{275807}a^{2}-\frac{22943}{275807}$, $\frac{14285}{2206456}a^{31}+\frac{111893}{35303296}a^{30}-\frac{201185}{8825824}a^{29}-\frac{230225}{17651648}a^{28}-\frac{59}{1103228}a^{27}+\frac{3571}{551614}a^{26}+\frac{14285}{53816}a^{25}+\frac{35710}{275807}a^{24}-\frac{249995}{275807}a^{23}-\frac{142840}{275807}a^{22}+\frac{3314237}{1103228}a^{21}+\frac{15265185}{8825824}a^{20}-\frac{3040}{275807}a^{19}-\frac{3966059}{4412912}a^{18}-\frac{159306419}{4412912}a^{17}-\frac{4856560}{275807}a^{16}+\frac{6628439}{53816}a^{15}+\frac{19411956}{275807}a^{14}-\frac{35535}{551614}a^{13}-\frac{9713120}{275807}a^{12}-\frac{46390913}{1103228}a^{11}-\frac{9883891}{551614}a^{10}-\frac{1169974}{275807}a^{9}-\frac{7081909}{1103228}a^{8}-\frac{7998855}{551614}a^{7}-\frac{2285440}{275807}a^{6}+\frac{10}{6727}a^{5}+\frac{1142720}{275807}a^{4}+\frac{44231}{8897}a^{3}+\frac{114272}{275807}a^{2}-\frac{19161}{39401}a-\frac{67079}{275807}$, $\frac{195025}{35303296}a^{31}+\frac{20793}{8825824}a^{30}-\frac{780099}{35303296}a^{29}-\frac{46367}{5043328}a^{28}+\frac{195025}{17651648}a^{27}+\frac{5771}{1260832}a^{26}+\frac{975125}{4412912}a^{25}+\frac{28855}{315208}a^{24}-\frac{975125}{1103228}a^{23}-\frac{28855}{78802}a^{22}+\frac{3315425}{1103228}a^{21}+\frac{1609113}{1260832}a^{20}-\frac{6435825}{4412912}a^{19}-\frac{196209}{315208}a^{18}-\frac{16577125}{551614}a^{17}-\frac{490535}{39401}a^{16}+\frac{265038975}{2206456}a^{15}+\frac{7842789}{157604}a^{14}-\frac{16577125}{275807}a^{13}-\frac{981070}{39401}a^{12}-\frac{6435825}{1103228}a^{11}+\frac{476093}{39401}a^{10}+\frac{6630850}{275807}a^{9}+\frac{231039}{157604}a^{8}-\frac{3900500}{275807}a^{7}-\frac{230840}{39401}a^{6}+\frac{1950250}{275807}a^{5}+\frac{115420}{39401}a^{4}+\frac{195025}{275807}a^{3}+\frac{11542}{39401}a^{2}-\frac{780100}{275807}a-\frac{323135}{275807}$, $\frac{5049}{2206456}a^{31}+\frac{12673}{17651648}a^{30}-\frac{161565}{17651648}a^{29}-\frac{4611}{1260832}a^{28}+\frac{20193}{4412912}a^{27}+\frac{81031}{17651648}a^{26}+\frac{25245}{275807}a^{25}+\frac{119741}{4412912}a^{24}-\frac{807837}{2206456}a^{23}-\frac{92215}{630416}a^{22}+\frac{10986385}{8825824}a^{21}+\frac{4555975}{8825824}a^{20}-\frac{1332733}{2206456}a^{19}-\frac{340199}{551614}a^{18}-\frac{54933119}{4412912}a^{17}-\frac{2327639}{630416}a^{16}+\frac{54891539}{1103228}a^{15}+\frac{5482189}{275807}a^{14}-\frac{54931727}{2206456}a^{13}-\frac{54904077}{2206456}a^{12}-\frac{1332937}{551614}a^{11}+\frac{613553}{78802}a^{10}+\frac{2746598}{275807}a^{9}+\frac{1090139}{275807}a^{8}-\frac{1615645}{275807}a^{7}-\frac{1290911}{551614}a^{6}+\frac{1615639}{551614}a^{5}+\frac{580}{39401}a^{4}-\frac{390048}{275807}a^{3}+\frac{218938}{275807}a^{2}-\frac{323129}{275807}a+\frac{146740}{275807}$ 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:  \( 549552630502.74054 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 549552630502.74054 \cdot 100}{30\cdot\sqrt{794071845499378503449051136000000000000000000000000}}\cr\approx \mathstrut & 0.383562403194620 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^32 - 4*x^30 + 2*x^28 + 40*x^26 - 160*x^24 + 544*x^22 - 264*x^20 - 5440*x^18 + 21744*x^16 - 10880*x^14 - 1056*x^12 + 4352*x^10 - 2560*x^8 + 1280*x^6 + 128*x^4 - 512*x^2 + 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 - 4*x^30 + 2*x^28 + 40*x^26 - 160*x^24 + 544*x^22 - 264*x^20 - 5440*x^18 + 21744*x^16 - 10880*x^14 - 1056*x^12 + 4352*x^10 - 2560*x^8 + 1280*x^6 + 128*x^4 - 512*x^2 + 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 - 4*x^30 + 2*x^28 + 40*x^26 - 160*x^24 + 544*x^22 - 264*x^20 - 5440*x^18 + 21744*x^16 - 10880*x^14 - 1056*x^12 + 4352*x^10 - 2560*x^8 + 1280*x^6 + 128*x^4 - 512*x^2 + 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 - 4*x^30 + 2*x^28 + 40*x^26 - 160*x^24 + 544*x^22 - 264*x^20 - 5440*x^18 + 21744*x^16 - 10880*x^14 - 1056*x^12 + 4352*x^10 - 2560*x^8 + 1280*x^6 + 128*x^4 - 512*x^2 + 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

$C_2\times C_4^2$ (as 32T36):

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 $C_2\times C_4^2$
Character table for $C_2\times C_4^2$

Intermediate fields

\(\Q(\sqrt{-6}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-30}) \), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\zeta_{16})^+\), 4.0.18432.2, \(\Q(\sqrt{-6}, \sqrt{10})\), \(\Q(\sqrt{5}, \sqrt{-6})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{-15})\), 4.4.51200.1, 4.0.460800.2, \(\Q(\sqrt{-3}, \sqrt{10})\), \(\Q(\sqrt{-3}, \sqrt{5})\), 4.4.2304000.2, 4.0.256000.4, 4.4.2304000.1, 4.0.256000.2, \(\Q(\zeta_{15})^+\), 4.0.8000.2, 4.4.72000.1, \(\Q(\zeta_{5})\), 8.0.339738624.2, 8.0.207360000.1, 8.0.212336640000.7, 8.8.2621440000.1, 8.0.212336640000.4, 8.0.212336640000.5, 8.0.212336640000.2, 8.0.5308416000000.8, 8.0.5308416000000.4, 8.0.5184000000.1, 8.0.5184000000.5, 8.8.5308416000000.1, 8.0.65536000000.1, 8.8.5184000000.1, 8.0.64000000.2, 8.0.5308416000000.5, 8.0.5308416000000.1, \(\Q(\zeta_{15})\), 8.0.5184000000.3, 16.0.45086848686489600000000.1, 16.0.28179280429056000000000000.2, 16.0.26873856000000000000.2, 16.16.28179280429056000000000000.2, 16.0.4294967296000000000000.1, 16.0.28179280429056000000000000.5, 16.0.28179280429056000000000000.4

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 ${\href{/padicField/7.4.0.1}{4} }^{8}$ ${\href{/padicField/11.4.0.1}{4} }^{8}$ ${\href{/padicField/13.4.0.1}{4} }^{8}$ ${\href{/padicField/17.4.0.1}{4} }^{8}$ ${\href{/padicField/19.4.0.1}{4} }^{8}$ ${\href{/padicField/23.4.0.1}{4} }^{8}$ ${\href{/padicField/29.4.0.1}{4} }^{8}$ ${\href{/padicField/31.1.0.1}{1} }^{32}$ ${\href{/padicField/37.4.0.1}{4} }^{8}$ ${\href{/padicField/41.2.0.1}{2} }^{16}$ ${\href{/padicField/43.4.0.1}{4} }^{8}$ ${\href{/padicField/47.4.0.1}{4} }^{8}$ ${\href{/padicField/53.4.0.1}{4} }^{8}$ ${\href{/padicField/59.4.0.1}{4} }^{8}$

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 Deg $16$$4$$4$$44$
Deg $16$$4$$4$$44$
\(3\) 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}$