Properties

Label 16.16.810...016.2
Degree $16$
Signature $(16, 0)$
Discriminant $8.103\times 10^{25}$
Root discriminant \(41.62\)
Ramified primes $2,3,23$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^4:Q_8$ (as 16T333)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^16 - 32*x^14 - 16*x^13 + 340*x^12 + 264*x^11 - 1448*x^10 - 1200*x^9 + 2746*x^8 + 1936*x^7 - 2456*x^6 - 1072*x^5 + 956*x^4 + 80*x^3 - 152*x^2 + 32*x - 2)
 
Copy content gp:K = bnfinit(y^16 - 32*y^14 - 16*y^13 + 340*y^12 + 264*y^11 - 1448*y^10 - 1200*y^9 + 2746*y^8 + 1936*y^7 - 2456*y^6 - 1072*y^5 + 956*y^4 + 80*y^3 - 152*y^2 + 32*y - 2, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 32*x^14 - 16*x^13 + 340*x^12 + 264*x^11 - 1448*x^10 - 1200*x^9 + 2746*x^8 + 1936*x^7 - 2456*x^6 - 1072*x^5 + 956*x^4 + 80*x^3 - 152*x^2 + 32*x - 2);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 32*x^14 - 16*x^13 + 340*x^12 + 264*x^11 - 1448*x^10 - 1200*x^9 + 2746*x^8 + 1936*x^7 - 2456*x^6 - 1072*x^5 + 956*x^4 + 80*x^3 - 152*x^2 + 32*x - 2)
 

\( x^{16} - 32 x^{14} - 16 x^{13} + 340 x^{12} + 264 x^{11} - 1448 x^{10} - 1200 x^{9} + 2746 x^{8} + \cdots - 2 \) Copy content Toggle raw display

Copy content comment:Defining polynomial
 
Copy content sage:K.defining_polynomial()
 
Copy content gp:K.pol
 
Copy content magma:DefiningPolynomial(K);
 
Copy content oscar:defining_polynomial(K)
 

Invariants

Degree:  $16$
Copy content comment:Degree over Q
 
Copy content sage:K.degree()
 
Copy content gp:poldegree(K.pol)
 
Copy content magma:Degree(K);
 
Copy content oscar:degree(K)
 
Signature:  $(16, 0)$
Copy content comment:Signature
 
Copy content sage:K.signature()
 
Copy content gp:K.sign
 
Copy content magma:Signature(K);
 
Copy content oscar:signature(K)
 
Discriminant:   \(81030865406861733858902016\) \(\medspace = 2^{58}\cdot 3^{12}\cdot 23^{2}\) Copy content Toggle raw display
Copy content comment:Discriminant
 
Copy content sage:K.disc()
 
Copy content gp:K.disc
 
Copy content magma:OK := Integers(K); Discriminant(OK);
 
Copy content oscar:OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(41.62\)
Copy content comment:Root discriminant
 
Copy content sage:(K.disc().abs())^(1./K.degree())
 
Copy content gp:abs(K.disc)^(1/poldegree(K.pol))
 
Copy content magma:Abs(Discriminant(OK))^(1/Degree(K));
 
Copy content oscar:OK = ring_of_integers(K); (1.0 * abs(discriminant(OK)))^(1/degree(K))
 
Galois root discriminant:  $2^{31/8}3^{3/4}23^{1/2}\approx 160.39694500305947$
Ramified primes:   \(2\), \(3\), \(23\) Copy content Toggle raw display
Copy content comment:Ramified primes
 
Copy content sage:K.disc().support()
 
Copy content gp:factor(abs(K.disc))[,1]~
 
Copy content magma:PrimeDivisors(Discriminant(OK));
 
Copy content oscar:prime_divisors(discriminant(OK))
 
Discriminant root field:  \(\Q\)
$\Aut(K/\Q)$:   $C_2^2$
Copy content comment:Automorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphism_group(K)
 
This field is not Galois over $\Q$.
This is not a CM field.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{94505624497}a^{15}-\frac{2195302511}{94505624497}a^{14}+\frac{46272346319}{94505624497}a^{13}-\frac{7385402586}{94505624497}a^{12}+\frac{33020976187}{94505624497}a^{11}-\frac{10747952432}{94505624497}a^{10}-\frac{46455214434}{94505624497}a^{9}+\frac{19103433847}{94505624497}a^{8}+\frac{11769730917}{94505624497}a^{7}+\frac{884492894}{94505624497}a^{6}+\frac{6262287673}{94505624497}a^{5}-\frac{33548283249}{94505624497}a^{4}-\frac{15888546193}{94505624497}a^{3}-\frac{21475212361}{94505624497}a^{2}-\frac{155730179}{94505624497}a+\frac{334286025}{94505624497}$ Copy content Toggle raw display

Copy content comment:Integral basis
 
Copy content sage:K.integral_basis()
 
Copy content gp:K.zk
 
Copy content magma:IntegralBasis(K);
 
Copy content oscar:basis(OK)
 

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

Ideal class group:  Trivial group, which has order $1$ (assuming GRH)
Copy content comment:Class group
 
Copy content sage:K.class_group().invariants()
 
Copy content gp:K.clgp
 
Copy content magma:ClassGroup(K);
 
Copy content oscar:class_group(K)
 
Narrow class group:  $C_{2}\times C_{2}$, which has order $4$ (assuming GRH)
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 

Unit group

Copy content comment:Unit group
 
Copy content sage:UK = K.unit_group()
 
Copy content magma:UK, fUK := UnitGroup(K);
 
Copy content oscar:UK, fUK = unit_group(OK)
 
Rank:  $15$
Copy content comment:Unit rank
 
Copy content sage:UK.rank()
 
Copy content gp:K.fu
 
Copy content magma:UnitRank(K);
 
Copy content oscar:rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
Copy content comment:Generator for roots of unity
 
Copy content sage:UK.torsion_generator()
 
Copy content gp:K.tu[2]
 
Copy content magma:K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Copy content oscar:torsion_units_generator(OK)
 
Fundamental units:   $\frac{770338611294}{94505624497}a^{15}+\frac{261254875094}{94505624497}a^{14}-\frac{24565233294000}{94505624497}a^{13}-\frac{20666809958125}{94505624497}a^{12}+\frac{255004038828550}{94505624497}a^{11}+\frac{290211288981954}{94505624497}a^{10}-\frac{10\cdots 92}{94505624497}a^{9}-\frac{12\cdots 69}{94505624497}a^{8}+\frac{16\cdots 52}{94505624497}a^{7}+\frac{20\cdots 92}{94505624497}a^{6}-\frac{11\cdots 16}{94505624497}a^{5}-\frac{12\cdots 33}{94505624497}a^{4}+\frac{310696101454696}{94505624497}a^{3}+\frac{175387545400706}{94505624497}a^{2}-\frac{56865285187904}{94505624497}a+\frac{4112597855779}{94505624497}$, $\frac{365842026544}{94505624497}a^{15}-\frac{93305434520}{94505624497}a^{14}+\frac{11685048147828}{94505624497}a^{13}+\frac{8832459932465}{94505624497}a^{12}-\frac{122191292321016}{94505624497}a^{11}-\frac{127743486619206}{94505624497}a^{10}+\frac{497733263236408}{94505624497}a^{9}+\frac{566147082744961}{94505624497}a^{8}-\frac{862399549122564}{94505624497}a^{7}-\frac{929380185670760}{94505624497}a^{6}+\frac{664830561290216}{94505624497}a^{5}+\frac{563822144241648}{94505624497}a^{4}-\frac{207840665973436}{94505624497}a^{3}-\frac{83655261278600}{94505624497}a^{2}+\frac{34397702858016}{94505624497}a-\frac{2770676906739}{94505624497}$, $\frac{2416122222960}{94505624497}a^{15}-\frac{682020502860}{94505624497}a^{14}+\frac{77131680745908}{94505624497}a^{13}+\frac{60427017754817}{94505624497}a^{12}-\frac{804667982591228}{94505624497}a^{11}-\frac{865031393930142}{94505624497}a^{10}+\frac{32\cdots 84}{94505624497}a^{9}+\frac{38\cdots 44}{94505624497}a^{8}-\frac{55\cdots 72}{94505624497}a^{7}-\frac{62\cdots 60}{94505624497}a^{6}+\frac{41\cdots 64}{94505624497}a^{5}+\frac{37\cdots 74}{94505624497}a^{4}-\frac{12\cdots 64}{94505624497}a^{3}-\frac{542184724046836}{94505624497}a^{2}+\frac{214479923533920}{94505624497}a-\frac{17380974740521}{94505624497}$, $\frac{2136512616113}{94505624497}a^{15}+\frac{643107498697}{94505624497}a^{14}-\frac{68174271411253}{94505624497}a^{13}-\frac{54713854396256}{94505624497}a^{12}+\frac{709943809492245}{94505624497}a^{11}+\frac{777972861484255}{94505624497}a^{10}-\frac{28\cdots 34}{94505624497}a^{9}-\frac{34\cdots 15}{94505624497}a^{8}+\frac{48\cdots 96}{94505624497}a^{7}+\frac{55\cdots 74}{94505624497}a^{6}-\frac{35\cdots 80}{94505624497}a^{5}-\frac{33\cdots 48}{94505624497}a^{4}+\frac{10\cdots 00}{94505624497}a^{3}+\frac{481999842268427}{94505624497}a^{2}-\frac{182682288143860}{94505624497}a+\frac{14711728224147}{94505624497}$, $\frac{2604204005162}{94505624497}a^{15}+\frac{732481936248}{94505624497}a^{14}-\frac{83117974535283}{94505624497}a^{13}-\frac{65050585276323}{94505624497}a^{12}+\frac{866784714220583}{94505624497}a^{11}+\frac{931303124242375}{94505624497}a^{10}-\frac{35\cdots 08}{94505624497}a^{9}-\frac{41\cdots 86}{94505624497}a^{8}+\frac{59\cdots 16}{94505624497}a^{7}+\frac{67\cdots 95}{94505624497}a^{6}-\frac{44\cdots 92}{94505624497}a^{5}-\frac{40\cdots 42}{94505624497}a^{4}+\frac{13\cdots 90}{94505624497}a^{3}+\frac{576760122841001}{94505624497}a^{2}-\frac{224794196611986}{94505624497}a+\frac{18031501333549}{94505624497}$, $\frac{1732016031363}{94505624497}a^{15}-\frac{475158058123}{94505624497}a^{14}+\frac{55294086265081}{94505624497}a^{13}+\frac{42879504370596}{94505624497}a^{12}-\frac{577131062984711}{94505624497}a^{11}-\frac{615505059121507}{94505624497}a^{10}+\frac{23\cdots 50}{94505624497}a^{9}+\frac{27\cdots 07}{94505624497}a^{8}-\frac{40\cdots 08}{94505624497}a^{7}-\frac{44\cdots 42}{94505624497}a^{6}+\frac{30\cdots 80}{94505624497}a^{5}+\frac{26\cdots 63}{94505624497}a^{4}-\frac{933666267576940}{94505624497}a^{3}-\frac{390267558146321}{94505624497}a^{2}+\frac{160309211438469}{94505624497}a-\frac{13180796026113}{94505624497}$, $\frac{2194625875361}{94505624497}a^{15}+\frac{676592613920}{94505624497}a^{14}-\frac{70014525744201}{94505624497}a^{13}-\frac{56689997947122}{94505624497}a^{12}+\frac{728541460540973}{94505624497}a^{11}+\frac{803635725256626}{94505624497}a^{10}-\frac{29\cdots 32}{94505624497}a^{9}-\frac{35\cdots 86}{94505624497}a^{8}+\frac{49\cdots 80}{94505624497}a^{7}+\frac{57\cdots 62}{94505624497}a^{6}-\frac{36\cdots 06}{94505624497}a^{5}-\frac{34\cdots 19}{94505624497}a^{4}+\frac{10\cdots 38}{94505624497}a^{3}+\frac{493579289316785}{94505624497}a^{2}-\frac{180888096193925}{94505624497}a+\frac{13910368898381}{94505624497}$, $\frac{5906505467797}{94505624497}a^{15}-\frac{1792840374820}{94505624497}a^{14}+\frac{188471462337482}{94505624497}a^{13}+\frac{151706408552215}{94505624497}a^{12}-\frac{19\cdots 24}{94505624497}a^{11}-\frac{21\cdots 90}{94505624497}a^{10}+\frac{79\cdots 00}{94505624497}a^{9}+\frac{94\cdots 84}{94505624497}a^{8}-\frac{13\cdots 66}{94505624497}a^{7}-\frac{15\cdots 45}{94505624497}a^{6}+\frac{98\cdots 76}{94505624497}a^{5}+\frac{93\cdots 66}{94505624497}a^{4}-\frac{28\cdots 94}{94505624497}a^{3}-\frac{13\cdots 34}{94505624497}a^{2}+\frac{496114585159086}{94505624497}a-\frac{39198590421981}{94505624497}$, $\frac{1018071934453}{94505624497}a^{15}-\frac{225893900141}{94505624497}a^{14}+\frac{32499793607142}{94505624497}a^{13}+\frac{23516292797386}{94505624497}a^{12}-\frac{340063311709684}{94505624497}a^{11}-\frac{344234391526548}{94505624497}a^{10}+\frac{13\cdots 27}{94505624497}a^{9}+\frac{15\cdots 15}{94505624497}a^{8}-\frac{24\cdots 76}{94505624497}a^{7}-\frac{24\cdots 77}{94505624497}a^{6}+\frac{18\cdots 05}{94505624497}a^{5}+\frac{14\cdots 22}{94505624497}a^{4}-\frac{610487525010724}{94505624497}a^{3}-\frac{215924050535052}{94505624497}a^{2}+\frac{100506636775859}{94505624497}a-\frac{8776246973779}{94505624497}$, $\frac{613307652362}{94505624497}a^{15}-\frac{186794527605}{94505624497}a^{14}+\frac{19558362310937}{94505624497}a^{13}+\frac{15766877042930}{94505624497}a^{12}-\frac{203385792665111}{94505624497}a^{11}-\frac{223604023523976}{94505624497}a^{10}+\frac{816484662630350}{94505624497}a^{9}+\frac{981155222113006}{94505624497}a^{8}-\frac{13\cdots 03}{94505624497}a^{7}-\frac{15\cdots 21}{94505624497}a^{6}+\frac{997801457027904}{94505624497}a^{5}+\frac{944422787257406}{94505624497}a^{4}-\frac{279321026168070}{94505624497}a^{3}-\frac{130302159953896}{94505624497}a^{2}+\frac{48319545965508}{94505624497}a-\frac{3833428168125}{94505624497}$, $\frac{3140904480274}{94505624497}a^{15}+\frac{891260784726}{94505624497}a^{14}-\frac{100242665551302}{94505624497}a^{13}-\frac{78699542385585}{94505624497}a^{12}+\frac{10\cdots 44}{94505624497}a^{11}+\frac{11\cdots 25}{94505624497}a^{10}-\frac{42\cdots 78}{94505624497}a^{9}-\frac{49\cdots 00}{94505624497}a^{8}+\frac{71\cdots 58}{94505624497}a^{7}+\frac{81\cdots 90}{94505624497}a^{6}-\frac{53\cdots 53}{94505624497}a^{5}-\frac{48\cdots 34}{94505624497}a^{4}+\frac{15\cdots 56}{94505624497}a^{3}+\frac{699142460872152}{94505624497}a^{2}-\frac{273598674065041}{94505624497}a+\frac{22040949018421}{94505624497}$, $\frac{15433824179}{94505624497}a^{15}-\frac{20821102238}{94505624497}a^{14}-\frac{503380251731}{94505624497}a^{13}+\frac{399318218365}{94505624497}a^{12}+\frac{5895652767127}{94505624497}a^{11}-\frac{2251865195841}{94505624497}a^{10}-\frac{31057091529070}{94505624497}a^{9}+\frac{3296887054949}{94505624497}a^{8}+\frac{78766624926249}{94505624497}a^{7}+\frac{4917939235580}{94505624497}a^{6}-\frac{92573303966500}{94505624497}a^{5}-\frac{11909952949435}{94505624497}a^{4}+\frac{44485369466300}{94505624497}a^{3}+\frac{5503532906488}{94505624497}a^{2}-\frac{5940008327786}{94505624497}a+\frac{388218145389}{94505624497}$, $\frac{640156228443}{94505624497}a^{15}+\frac{198159298265}{94505624497}a^{14}-\frac{20398980750376}{94505624497}a^{13}-\frac{16575324027875}{94505624497}a^{12}+\frac{211775143172404}{94505624497}a^{11}+\frac{234700171104758}{94505624497}a^{10}-\frac{846874859896990}{94505624497}a^{9}-\frac{10\cdots 35}{94505624497}a^{8}+\frac{14\cdots 96}{94505624497}a^{7}+\frac{16\cdots 69}{94505624497}a^{6}-\frac{10\cdots 76}{94505624497}a^{5}-\frac{996721925807738}{94505624497}a^{4}+\frac{275009193513134}{94505624497}a^{3}+\frac{141701039572956}{94505624497}a^{2}-\frac{47312043404038}{94505624497}a+\frac{3239117016641}{94505624497}$, $\frac{1438726354224}{94505624497}a^{15}+\frac{458504439848}{94505624497}a^{14}-\frac{45905277463333}{94505624497}a^{13}-\frac{37635736203978}{94505624497}a^{12}+\frac{477552249152343}{94505624497}a^{11}+\frac{531808909917042}{94505624497}a^{10}-\frac{19\cdots 17}{94505624497}a^{9}-\frac{23\cdots 48}{94505624497}a^{8}+\frac{32\cdots 18}{94505624497}a^{7}+\frac{38\cdots 95}{94505624497}a^{6}-\frac{23\cdots 64}{94505624497}a^{5}-\frac{22\cdots 01}{94505624497}a^{4}+\frac{671899822823408}{94505624497}a^{3}+\frac{331301244436585}{94505624497}a^{2}-\frac{118826819569234}{94505624497}a+\frac{9159028954731}{94505624497}$, $\frac{2329987138950}{94505624497}a^{15}-\frac{724782257314}{94505624497}a^{14}+\frac{74350348164534}{94505624497}a^{13}+\frac{60390779028594}{94505624497}a^{12}-\frac{773923102612232}{94505624497}a^{11}-\frac{855601102560116}{94505624497}a^{10}+\frac{31\cdots 17}{94505624497}a^{9}+\frac{37\cdots 94}{94505624497}a^{8}-\frac{52\cdots 44}{94505624497}a^{7}-\frac{61\cdots 86}{94505624497}a^{6}+\frac{38\cdots 70}{94505624497}a^{5}+\frac{37\cdots 89}{94505624497}a^{4}-\frac{11\cdots 40}{94505624497}a^{3}-\frac{535120051373192}{94505624497}a^{2}+\frac{197200308295084}{94505624497}a-\frac{15440837915279}{94505624497}$ Copy content Toggle raw display (assuming GRH)
Copy content comment:Fundamental units
 
Copy content sage:UK.fundamental_units()
 
Copy content gp:K.fu
 
Copy content magma:[K|fUK(g): g in Generators(UK)];
 
Copy content oscar:[K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 97751967.49566491 \) (assuming GRH)
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 
Unit signature rank:  \( 14 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{16}\cdot(2\pi)^{0}\cdot 97751967.49566491 \cdot 1}{2\cdot\sqrt{81030865406861733858902016}}\cr\approx \mathstrut & 0.355836262291574 \end{aligned}\] (assuming GRH)

Copy content comment:Analytic class number formula
 
Copy content sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^16 - 32*x^14 - 16*x^13 + 340*x^12 + 264*x^11 - 1448*x^10 - 1200*x^9 + 2746*x^8 + 1936*x^7 - 2456*x^6 - 1072*x^5 + 956*x^4 + 80*x^3 - 152*x^2 + 32*x - 2) 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))))
 
Copy content gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^16 - 32*x^14 - 16*x^13 + 340*x^12 + 264*x^11 - 1448*x^10 - 1200*x^9 + 2746*x^8 + 1936*x^7 - 2456*x^6 - 1072*x^5 + 956*x^4 + 80*x^3 - 152*x^2 + 32*x - 2, 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)))]
 
Copy content magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^16 - 32*x^14 - 16*x^13 + 340*x^12 + 264*x^11 - 1448*x^10 - 1200*x^9 + 2746*x^8 + 1936*x^7 - 2456*x^6 - 1072*x^5 + 956*x^4 + 80*x^3 - 152*x^2 + 32*x - 2); 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)));
 
Copy content oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = polynomial_ring(QQ); K, a = number_field(x^16 - 32*x^14 - 16*x^13 + 340*x^12 + 264*x^11 - 1448*x^10 - 1200*x^9 + 2746*x^8 + 1936*x^7 - 2456*x^6 - 1072*x^5 + 956*x^4 + 80*x^3 - 152*x^2 + 32*x - 2); 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

$C_2^4:Q_8$ (as 16T333):

Copy content comment:Galois group
 
Copy content sage:K.galois_group()
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
A solvable group of order 128
The 26 conjugacy class representatives for $C_2^4:Q_8$
Character table for $C_2^4:Q_8$

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{6}) \), \(\Q(\zeta_{24})^+\), 8.8.562607161344.2, 8.8.12230590464.1, 8.8.62511906816.1

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

Copy content comment:Intermediate fields
 
Copy content sage:K.subfields()[1:-1]
 
Copy content gp:L = nfsubfields(K); L[2..length(L)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Degree 16 siblings: data not computed
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 ${\href{/padicField/5.4.0.1}{4} }^{4}$ ${\href{/padicField/7.4.0.1}{4} }^{4}$ ${\href{/padicField/11.4.0.1}{4} }^{4}$ ${\href{/padicField/13.4.0.1}{4} }^{4}$ ${\href{/padicField/17.4.0.1}{4} }^{4}$ ${\href{/padicField/19.4.0.1}{4} }^{4}$ R ${\href{/padicField/29.4.0.1}{4} }^{4}$ ${\href{/padicField/31.4.0.1}{4} }^{4}$ ${\href{/padicField/37.4.0.1}{4} }^{4}$ ${\href{/padicField/41.4.0.1}{4} }^{4}$ ${\href{/padicField/43.4.0.1}{4} }^{4}$ ${\href{/padicField/47.2.0.1}{2} }^{8}$ ${\href{/padicField/53.4.0.1}{4} }^{4}$ ${\href{/padicField/59.4.0.1}{4} }^{4}$

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

Copy content comment:Frobenius cycle types
 
Copy content sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
Copy content gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
Copy content magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
Copy content oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_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 2.1.16.58n1.779$x^{16} + 8 x^{15} + 8 x^{14} + 4 x^{12} + 8 x^{11} + 2 x^{8} + 8 x^{6} + 20 x^{4} + 16 x^{3} + 8 x^{2} + 16 x + 14$$16$$1$$58$16T333$$[2, 3, 3, \frac{7}{2}, 4, \frac{17}{4}]^{2}$$
\(3\) Copy content Toggle raw display 3.4.4.12a1.3$x^{16} + 8 x^{15} + 24 x^{14} + 32 x^{13} + 24 x^{12} + 48 x^{11} + 96 x^{10} + 64 x^{9} + 24 x^{8} + 96 x^{7} + 96 x^{6} + 32 x^{4} + 64 x^{3} + 19$$4$$4$$12$$C_4:C_4$$$[\ ]_{4}^{4}$$
\(23\) Copy content Toggle raw display $\Q_{23}$$x + 18$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{23}$$x + 18$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{23}$$x + 18$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{23}$$x + 18$$1$$1$$0$Trivial$$[\ ]$$
23.2.1.0a1.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$$[\ ]^{2}$$
23.1.2.1a1.2$x^{2} + 115$$2$$1$$1$$C_2$$$[\ ]_{2}$$
23.2.1.0a1.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$$[\ ]^{2}$$
23.2.1.0a1.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$$[\ ]^{2}$$
23.1.2.1a1.2$x^{2} + 115$$2$$1$$1$$C_2$$$[\ ]_{2}$$
23.2.1.0a1.1$x^{2} + 21 x + 5$$1$$2$$0$$C_2$$$[\ ]^{2}$$

Spectrum of ring of integers

(0)(0)(2)(3)(5)(7)(11)(13)(17)(19)(23)(29)(31)(37)(41)(43)(47)(53)(59)