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Properties

Label 32.0.184...576.1
Degree $32$
Signature $[0, 16]$
Discriminant $1.841\times 10^{50}$
Root discriminant \(37.22\)
Ramified primes $2,3,7,17$
Class number not computed
Class group not computed
Galois group $D_4\times C_2^3$ (as 32T273)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 28*x^30 + 502*x^28 - 5304*x^26 + 40319*x^24 - 205348*x^22 + 769870*x^20 - 2005384*x^18 + 3886261*x^16 - 5310988*x^14 + 5324752*x^12 - 3494768*x^10 + 1644300*x^8 - 452512*x^6 + 81152*x^4 - 1184*x^2 + 16)
 
gp: K = bnfinit(y^32 - 28*y^30 + 502*y^28 - 5304*y^26 + 40319*y^24 - 205348*y^22 + 769870*y^20 - 2005384*y^18 + 3886261*y^16 - 5310988*y^14 + 5324752*y^12 - 3494768*y^10 + 1644300*y^8 - 452512*y^6 + 81152*y^4 - 1184*y^2 + 16, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 - 28*x^30 + 502*x^28 - 5304*x^26 + 40319*x^24 - 205348*x^22 + 769870*x^20 - 2005384*x^18 + 3886261*x^16 - 5310988*x^14 + 5324752*x^12 - 3494768*x^10 + 1644300*x^8 - 452512*x^6 + 81152*x^4 - 1184*x^2 + 16);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^32 - 28*x^30 + 502*x^28 - 5304*x^26 + 40319*x^24 - 205348*x^22 + 769870*x^20 - 2005384*x^18 + 3886261*x^16 - 5310988*x^14 + 5324752*x^12 - 3494768*x^10 + 1644300*x^8 - 452512*x^6 + 81152*x^4 - 1184*x^2 + 16)
 

\( x^{32} - 28 x^{30} + 502 x^{28} - 5304 x^{26} + 40319 x^{24} - 205348 x^{22} + 769870 x^{20} - 2005384 x^{18} + \cdots + 16 \) 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:   \(184085596562491198850839682056048266622946116632576\) \(\medspace = 2^{64}\cdot 3^{16}\cdot 7^{16}\cdot 17^{8}\) 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:  \(37.22\)
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:  $2^{2}3^{1/2}7^{1/2}17^{1/2}\approx 75.57777451076474$
Ramified primes:   \(2\), \(3\), \(7\), \(17\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{8}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{9}-\frac{1}{2}a^{5}$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{10}-\frac{1}{2}a^{6}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{11}-\frac{1}{2}a^{7}$, $\frac{1}{16}a^{16}-\frac{1}{4}a^{15}+\frac{1}{8}a^{14}+\frac{1}{16}a^{12}-\frac{1}{4}a^{11}+\frac{1}{8}a^{10}-\frac{1}{2}a^{9}-\frac{5}{16}a^{8}+\frac{1}{4}a^{7}+\frac{3}{8}a^{6}-\frac{1}{2}a^{5}+\frac{3}{8}a^{4}-\frac{1}{2}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a+\frac{1}{4}$, $\frac{1}{16}a^{17}+\frac{1}{8}a^{15}+\frac{1}{16}a^{13}+\frac{1}{8}a^{11}-\frac{1}{2}a^{10}-\frac{5}{16}a^{9}+\frac{3}{8}a^{7}-\frac{1}{2}a^{6}+\frac{3}{8}a^{5}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}+\frac{1}{4}a$, $\frac{1}{16}a^{18}-\frac{3}{16}a^{14}-\frac{1}{2}a^{11}+\frac{7}{16}a^{10}-\frac{1}{2}a^{7}-\frac{3}{8}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}$, $\frac{1}{32}a^{19}+\frac{5}{32}a^{15}-\frac{1}{4}a^{12}+\frac{15}{32}a^{11}-\frac{1}{2}a^{9}+\frac{1}{4}a^{8}-\frac{7}{16}a^{7}-\frac{1}{2}a^{6}+\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{8}a^{3}-\frac{1}{4}a$, $\frac{1}{32}a^{20}-\frac{1}{32}a^{16}-\frac{1}{4}a^{15}+\frac{1}{8}a^{14}-\frac{1}{4}a^{13}-\frac{7}{32}a^{12}-\frac{1}{4}a^{11}-\frac{3}{8}a^{10}-\frac{1}{4}a^{9}-\frac{1}{4}a^{7}-\frac{3}{8}a^{6}+\frac{1}{4}a^{5}+\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a+\frac{1}{4}$, $\frac{1}{32}a^{21}-\frac{1}{32}a^{17}+\frac{1}{8}a^{15}-\frac{1}{4}a^{14}-\frac{7}{32}a^{13}-\frac{3}{8}a^{11}-\frac{1}{4}a^{10}-\frac{1}{2}a^{8}-\frac{3}{8}a^{7}+\frac{1}{4}a^{6}+\frac{1}{4}a^{5}-\frac{1}{2}a^{2}+\frac{1}{4}a$, $\frac{1}{32}a^{22}-\frac{1}{32}a^{18}-\frac{1}{4}a^{15}+\frac{1}{32}a^{14}-\frac{1}{4}a^{11}+\frac{1}{4}a^{10}-\frac{1}{2}a^{9}-\frac{1}{4}a^{8}+\frac{1}{4}a^{7}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}$, $\frac{1}{32}a^{23}+\frac{3}{16}a^{15}-\frac{1}{4}a^{12}-\frac{9}{32}a^{11}-\frac{1}{2}a^{10}+\frac{1}{4}a^{9}+\frac{1}{4}a^{8}-\frac{7}{16}a^{7}-\frac{1}{2}a^{6}-\frac{1}{4}a^{4}-\frac{3}{8}a^{3}+\frac{1}{4}a$, $\frac{1}{160}a^{24}+\frac{1}{80}a^{22}-\frac{1}{160}a^{20}+\frac{3}{160}a^{16}-\frac{1}{4}a^{15}-\frac{1}{10}a^{14}+\frac{1}{16}a^{12}-\frac{1}{4}a^{11}+\frac{37}{80}a^{10}-\frac{1}{2}a^{9}-\frac{13}{80}a^{8}-\frac{1}{4}a^{7}-\frac{7}{20}a^{6}-\frac{1}{2}a^{5}-\frac{3}{8}a^{4}+\frac{2}{5}a^{2}-\frac{1}{2}a+\frac{3}{20}$, $\frac{1}{160}a^{25}+\frac{1}{80}a^{23}-\frac{1}{160}a^{21}+\frac{3}{160}a^{17}-\frac{1}{10}a^{15}+\frac{1}{16}a^{13}+\frac{37}{80}a^{11}-\frac{1}{2}a^{10}-\frac{13}{80}a^{9}-\frac{1}{2}a^{8}-\frac{7}{20}a^{7}-\frac{1}{2}a^{6}-\frac{3}{8}a^{5}-\frac{1}{2}a^{4}+\frac{2}{5}a^{3}-\frac{1}{2}a^{2}+\frac{3}{20}a$, $\frac{1}{160}a^{26}+\frac{1}{80}a^{20}-\frac{1}{80}a^{18}-\frac{1}{80}a^{16}-\frac{1}{4}a^{15}+\frac{7}{160}a^{14}-\frac{3}{80}a^{12}+\frac{1}{4}a^{11}-\frac{7}{80}a^{10}-\frac{2}{5}a^{8}-\frac{1}{4}a^{7}-\frac{17}{40}a^{6}-\frac{1}{2}a^{5}+\frac{3}{20}a^{4}-\frac{2}{5}a^{2}-\frac{3}{10}$, $\frac{1}{160}a^{27}+\frac{1}{80}a^{21}-\frac{1}{80}a^{19}-\frac{1}{80}a^{17}+\frac{7}{160}a^{15}-\frac{3}{80}a^{13}-\frac{7}{80}a^{11}-\frac{2}{5}a^{9}-\frac{17}{40}a^{7}-\frac{1}{2}a^{6}+\frac{3}{20}a^{5}-\frac{2}{5}a^{3}-\frac{3}{10}a$, $\frac{1}{175840}a^{28}+\frac{1}{1099}a^{26}-\frac{13}{43960}a^{24}-\frac{8}{785}a^{22}+\frac{67}{17584}a^{20}+\frac{409}{87920}a^{18}-\frac{2659}{175840}a^{16}-\frac{1}{4}a^{15}-\frac{8177}{87920}a^{14}+\frac{9809}{43960}a^{12}+\frac{1}{4}a^{11}-\frac{35291}{87920}a^{10}-\frac{1}{2}a^{9}+\frac{15287}{87920}a^{8}+\frac{1}{4}a^{7}+\frac{389}{785}a^{6}-\frac{1}{2}a^{5}+\frac{1429}{43960}a^{4}-\frac{851}{10990}a^{2}-\frac{1}{2}a+\frac{8569}{21980}$, $\frac{1}{175840}a^{29}+\frac{1}{1099}a^{27}-\frac{13}{43960}a^{25}-\frac{8}{785}a^{23}+\frac{67}{17584}a^{21}+\frac{409}{87920}a^{19}-\frac{2659}{175840}a^{17}-\frac{8177}{87920}a^{15}+\frac{9809}{43960}a^{13}-\frac{35291}{87920}a^{11}-\frac{1}{2}a^{10}+\frac{15287}{87920}a^{9}-\frac{1}{2}a^{8}+\frac{389}{785}a^{7}-\frac{1}{2}a^{6}+\frac{1429}{43960}a^{5}-\frac{851}{10990}a^{3}-\frac{1}{2}a^{2}+\frac{8569}{21980}a$, $\frac{1}{19\!\cdots\!60}a^{30}-\frac{34\!\cdots\!41}{19\!\cdots\!60}a^{28}+\frac{51\!\cdots\!25}{19\!\cdots\!76}a^{26}-\frac{47\!\cdots\!67}{19\!\cdots\!60}a^{24}-\frac{32\!\cdots\!11}{95\!\cdots\!80}a^{22}-\frac{28\!\cdots\!91}{19\!\cdots\!60}a^{20}-\frac{56\!\cdots\!57}{22\!\cdots\!20}a^{18}+\frac{17\!\cdots\!85}{40\!\cdots\!76}a^{16}-\frac{1}{4}a^{15}-\frac{16\!\cdots\!17}{95\!\cdots\!80}a^{14}+\frac{44\!\cdots\!88}{70\!\cdots\!83}a^{12}-\frac{1}{4}a^{11}-\frac{38\!\cdots\!37}{19\!\cdots\!76}a^{10}+\frac{23\!\cdots\!13}{95\!\cdots\!80}a^{8}+\frac{1}{4}a^{7}+\frac{48\!\cdots\!01}{11\!\cdots\!10}a^{6}-\frac{1}{2}a^{5}+\frac{31\!\cdots\!47}{68\!\cdots\!20}a^{4}+\frac{49\!\cdots\!66}{59\!\cdots\!55}a^{2}-\frac{1}{2}a+\frac{89\!\cdots\!23}{23\!\cdots\!20}$, $\frac{1}{38\!\cdots\!20}a^{31}+\frac{18\!\cdots\!87}{95\!\cdots\!80}a^{29}+\frac{98\!\cdots\!07}{54\!\cdots\!36}a^{27}+\frac{14\!\cdots\!77}{47\!\cdots\!40}a^{25}-\frac{55\!\cdots\!43}{38\!\cdots\!20}a^{23}-\frac{83\!\cdots\!43}{95\!\cdots\!80}a^{21}+\frac{12\!\cdots\!61}{22\!\cdots\!20}a^{19}-\frac{76\!\cdots\!23}{28\!\cdots\!20}a^{17}-\frac{73\!\cdots\!73}{54\!\cdots\!60}a^{15}-\frac{1}{4}a^{14}+\frac{80\!\cdots\!73}{56\!\cdots\!40}a^{13}-\frac{1}{4}a^{12}-\frac{15\!\cdots\!67}{19\!\cdots\!60}a^{11}+\frac{1}{4}a^{10}-\frac{44\!\cdots\!43}{47\!\cdots\!40}a^{9}-\frac{1}{4}a^{8}-\frac{15\!\cdots\!59}{95\!\cdots\!80}a^{7}-\frac{1}{4}a^{6}+\frac{34\!\cdots\!09}{95\!\cdots\!88}a^{5}+\frac{1}{4}a^{4}+\frac{48\!\cdots\!37}{23\!\cdots\!20}a^{3}-\frac{1}{2}a^{2}-\frac{49\!\cdots\!97}{11\!\cdots\!10}a-\frac{1}{2}$ 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{514996232529175576668093054984868617}{95394168824855907822536829440893888880} a^{31} + \frac{988106520635483486035703712791161}{1362773840355084397464811849155626984} a^{30} - \frac{3604382449022270539473796127595209947}{23848542206213976955634207360223472220} a^{29} - \frac{96823285094289232263717895106937049}{4769708441242795391126841472044694444} a^{28} + \frac{258462223445647671992516308044925698831}{95394168824855907822536829440893888880} a^{27} + \frac{13886130096013375056443394420900028831}{38157667529942363129014731776357555552} a^{26} - \frac{2730362074569507354458069031754085364851}{95394168824855907822536829440893888880} a^{25} - \frac{146695531565362803444888021738262891657}{38157667529942363129014731776357555552} a^{24} + \frac{4150351026456753812894507152519997081587}{19078833764971181564507365888178777776} a^{23} + \frac{39819189815535103881481029638740300655}{1362773840355084397464811849155626984} a^{22} - \frac{52829924586599207128709148866690083750073}{47697084412427953911268414720446944440} a^{21} - \frac{5676793647105181999500713932237637857833}{38157667529942363129014731776357555552} a^{20} + \frac{4771180308057980188798357914271404306931}{1149327335239227805090805173986673360} a^{19} + \frac{8009974033143960501828165962871708212}{14366591690490347563635064674833417} a^{18} - \frac{12129576781381995809678117950782616810659}{1122284339115951856735727405186986928} a^{17} - \frac{3257615084062207937618450952890564871835}{2244568678231903713471454810373973856} a^{16} + \frac{1996934928789693615039128479885794027651887}{95394168824855907822536829440893888880} a^{15} + \frac{107217631326153839452799035893432493557651}{38157667529942363129014731776357555552} a^{14} - \frac{80196186032143145663190025351933459682693}{2805710847789879641839318512967467320} a^{13} - \frac{4302705717676639163954075810640278224847}{1122284339115951856735727405186986928} a^{12} + \frac{390162646975655668015937163945657317473881}{13627738403550843974648118491556269840} a^{11} + \frac{18287419839542968524085243536523061079151}{4769708441242795391126841472044694444} a^{10} - \frac{1789268649131026700171573847451747328813087}{95394168824855907822536829440893888880} a^{9} - \frac{23885063909469021611976909738013364013113}{9539416882485590782253682944089388888} a^{8} + \frac{105086668555474653712982687262953750190429}{11924271103106988477817103680111736110} a^{7} + \frac{397654790784460289549565150933093256049}{340693460088771099366202962288906746} a^{6} - \frac{23052120519913087963677428519103333252099}{9539416882485590782253682944089388888} a^{5} - \frac{1500435983134946677055644497324633547337}{4769708441242795391126841472044694444} a^{4} + \frac{1035161254463375616980583511991251656371}{2384854220621397695563420736022347222} a^{3} + \frac{252139166040173027919746581252542158743}{4769708441242795391126841472044694444} a^{2} - \frac{151027767921222955513495610139025474777}{23848542206213976955634207360223472220} a - \frac{152300882045682185285006785323835843}{2384854220621397695563420736022347222} \)  (order $24$) 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 - 28*x^30 + 502*x^28 - 5304*x^26 + 40319*x^24 - 205348*x^22 + 769870*x^20 - 2005384*x^18 + 3886261*x^16 - 5310988*x^14 + 5324752*x^12 - 3494768*x^10 + 1644300*x^8 - 452512*x^6 + 81152*x^4 - 1184*x^2 + 16)
 
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 - 28*x^30 + 502*x^28 - 5304*x^26 + 40319*x^24 - 205348*x^22 + 769870*x^20 - 2005384*x^18 + 3886261*x^16 - 5310988*x^14 + 5324752*x^12 - 3494768*x^10 + 1644300*x^8 - 452512*x^6 + 81152*x^4 - 1184*x^2 + 16, 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 - 28*x^30 + 502*x^28 - 5304*x^26 + 40319*x^24 - 205348*x^22 + 769870*x^20 - 2005384*x^18 + 3886261*x^16 - 5310988*x^14 + 5324752*x^12 - 3494768*x^10 + 1644300*x^8 - 452512*x^6 + 81152*x^4 - 1184*x^2 + 16);
 
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 - 28*x^30 + 502*x^28 - 5304*x^26 + 40319*x^24 - 205348*x^22 + 769870*x^20 - 2005384*x^18 + 3886261*x^16 - 5310988*x^14 + 5324752*x^12 - 3494768*x^10 + 1644300*x^8 - 452512*x^6 + 81152*x^4 - 1184*x^2 + 16);
 
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

$D_4\times C_2^3$ (as 32T273):

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 $D_4\times C_2^3$
Character table for $D_4\times C_2^3$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{42}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-42}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{-21}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-14}) \), \(\Q(\sqrt{14}) \), 4.4.9792.1, 4.4.53312.1, 4.4.4352.1, 4.4.1919232.1, 4.0.1088.2, 4.0.479808.2, 4.0.39168.3, 4.0.213248.3, \(\Q(\sqrt{6}, \sqrt{14})\), \(\Q(\sqrt{-6}, \sqrt{-14})\), \(\Q(i, \sqrt{21})\), \(\Q(i, \sqrt{6})\), \(\Q(\sqrt{6}, \sqrt{-14})\), \(\Q(\sqrt{-6}, \sqrt{14})\), \(\Q(i, \sqrt{14})\), \(\Q(\sqrt{2}, \sqrt{21})\), \(\Q(\sqrt{3}, \sqrt{7})\), \(\Q(\sqrt{-3}, \sqrt{-7})\), \(\Q(\sqrt{-2}, \sqrt{21})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\sqrt{6}, \sqrt{7})\), \(\Q(\sqrt{-2}, \sqrt{-3})\), \(\Q(\sqrt{6}, \sqrt{-7})\), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\sqrt{-6}, \sqrt{-7})\), \(\Q(\sqrt{-2}, \sqrt{3})\), \(\Q(\sqrt{-6}, \sqrt{7})\), \(\Q(\sqrt{2}, \sqrt{-21})\), \(\Q(\sqrt{-2}, \sqrt{-21})\), \(\Q(\sqrt{3}, \sqrt{-7})\), \(\Q(\sqrt{-3}, \sqrt{7})\), \(\Q(\zeta_{8})\), \(\Q(i, \sqrt{42})\), \(\Q(\zeta_{12})\), \(\Q(i, \sqrt{7})\), \(\Q(\sqrt{2}, \sqrt{-7})\), \(\Q(\sqrt{-3}, \sqrt{-14})\), \(\Q(\sqrt{3}, \sqrt{-14})\), \(\Q(\sqrt{-2}, \sqrt{7})\), \(\Q(\sqrt{2}, \sqrt{7})\), \(\Q(\sqrt{3}, \sqrt{14})\), \(\Q(\sqrt{-3}, \sqrt{14})\), \(\Q(\sqrt{-2}, \sqrt{-7})\), 8.0.12745506816.1, 8.8.12745506816.1, 8.0.796594176.1, 8.0.796594176.2, 8.0.12745506816.3, 8.0.12745506816.8, 8.0.49787136.1, \(\Q(\zeta_{24})\), 8.0.12745506816.7, 8.0.12745506816.6, 8.0.12745506816.5, 8.0.12745506816.4, 8.0.12745506816.2, 8.0.157351936.1, 8.0.12745506816.9, 8.8.230215716864.1, 8.8.3683451469824.3, 8.0.230215716864.18, 8.0.3683451469824.25, 8.8.1534132224.1, 8.8.3683451469824.4, 8.0.1534132224.4, 8.0.3683451469824.32, 8.0.95883264.1, 8.0.230215716864.17, 8.0.1534132224.10, 8.0.3683451469824.82, 8.0.3683451469824.39, 8.0.3683451469824.20, 8.0.3683451469824.74, 8.0.3683451469824.21, 8.0.18939904.2, 8.0.3683451469824.60, 8.0.1534132224.8, 8.0.45474709504.3, 8.0.2842169344.2, 8.0.230215716864.9, 8.0.3683451469824.5, 8.0.45474709504.20, 8.8.3683451469824.5, 8.8.45474709504.1, 8.0.45474709504.2, 8.0.3683451469824.8, 16.0.162447943996702457856.1, 16.16.13567814730548585982590976.1, 16.0.13567814730548585982590976.4, 16.0.52999276291205413994496.1, 16.0.13567814730548585982590976.2, 16.0.13567814730548585982590976.10, 16.0.13567814730548585982590976.3, 16.0.2353561680715186176.2, 16.0.13567814730548585982590976.9, 16.0.13567814730548585982590976.5, 16.0.13567814730548585982590976.6, 16.0.13567814730548585982590976.8, 16.0.13567814730548585982590976.1, 16.0.13567814730548585982590976.7, 16.0.2067949204473187926016.1

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 ${\href{/padicField/5.4.0.1}{4} }^{8}$ R ${\href{/padicField/11.4.0.1}{4} }^{8}$ ${\href{/padicField/13.2.0.1}{2} }^{16}$ R ${\href{/padicField/19.2.0.1}{2} }^{16}$ ${\href{/padicField/23.2.0.1}{2} }^{16}$ ${\href{/padicField/29.4.0.1}{4} }^{8}$ ${\href{/padicField/31.2.0.1}{2} }^{16}$ ${\href{/padicField/37.4.0.1}{4} }^{8}$ ${\href{/padicField/41.2.0.1}{2} }^{16}$ ${\href{/padicField/43.2.0.1}{2} }^{16}$ ${\href{/padicField/47.2.0.1}{2} }^{16}$ ${\href{/padicField/53.2.0.1}{2} }^{16}$ ${\href{/padicField/59.2.0.1}{2} }^{16}$

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 2.8.16.6$x^{8} + 4 x^{7} + 24 x^{6} + 48 x^{5} + 56 x^{4} + 56 x^{3} + 64 x^{2} + 48 x + 36$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
2.8.16.6$x^{8} + 4 x^{7} + 24 x^{6} + 48 x^{5} + 56 x^{4} + 56 x^{3} + 64 x^{2} + 48 x + 36$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
2.8.16.6$x^{8} + 4 x^{7} + 24 x^{6} + 48 x^{5} + 56 x^{4} + 56 x^{3} + 64 x^{2} + 48 x + 36$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
2.8.16.6$x^{8} + 4 x^{7} + 24 x^{6} + 48 x^{5} + 56 x^{4} + 56 x^{3} + 64 x^{2} + 48 x + 36$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
\(3\) 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}$
\(7\) Copy content Toggle raw display 7.4.2.1$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
\(17\) Copy content Toggle raw display 17.2.0.1$x^{2} + 16 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} + 16 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} + 16 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} + 16 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} + 16 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} + 16 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} + 16 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} + 16 x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.4.2.1$x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 338 x^{3} + 31049 x^{2} + 420472 x + 123735$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$