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Properties

Label 17.1.604...088.1
Degree $17$
Signature $[1, 8]$
Discriminant $6.045\times 10^{23}$
Root discriminant \(25.06\)
Ramified prime $2$
Class number $1$
Class group trivial
Galois group $F_{17}$ (as 17T5)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^17 - 2*x^16 + 8*x^13 + 16*x^12 - 16*x^11 + 64*x^9 - 32*x^8 - 80*x^7 + 32*x^6 + 40*x^5 + 80*x^4 + 16*x^3 - 128*x^2 - 2*x + 68)
 
gp: K = bnfinit(y^17 - 2*y^16 + 8*y^13 + 16*y^12 - 16*y^11 + 64*y^9 - 32*y^8 - 80*y^7 + 32*y^6 + 40*y^5 + 80*y^4 + 16*y^3 - 128*y^2 - 2*y + 68, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^17 - 2*x^16 + 8*x^13 + 16*x^12 - 16*x^11 + 64*x^9 - 32*x^8 - 80*x^7 + 32*x^6 + 40*x^5 + 80*x^4 + 16*x^3 - 128*x^2 - 2*x + 68);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^17 - 2*x^16 + 8*x^13 + 16*x^12 - 16*x^11 + 64*x^9 - 32*x^8 - 80*x^7 + 32*x^6 + 40*x^5 + 80*x^4 + 16*x^3 - 128*x^2 - 2*x + 68)
 

\( x^{17} - 2 x^{16} + 8 x^{13} + 16 x^{12} - 16 x^{11} + 64 x^{9} - 32 x^{8} - 80 x^{7} + 32 x^{6} + \cdots + 68 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $17$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[1, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(604462909807314587353088\) \(\medspace = 2^{79}\) 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:  \(25.06\)
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:  $2^{79/16}\approx 30.64330498235436$
Ramified primes:   \(2\) 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:  \(\Q(\sqrt{2}) \)
$\card{ \Aut(K/\Q) }$:  $1$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{23\!\cdots\!82}a^{16}+\frac{76599019108737}{11\!\cdots\!91}a^{15}-\frac{31\!\cdots\!69}{11\!\cdots\!91}a^{14}-\frac{37\!\cdots\!82}{11\!\cdots\!91}a^{13}+\frac{54\!\cdots\!14}{11\!\cdots\!91}a^{12}+\frac{45\!\cdots\!51}{11\!\cdots\!91}a^{11}+\frac{44\!\cdots\!93}{11\!\cdots\!91}a^{10}+\frac{51\!\cdots\!16}{11\!\cdots\!91}a^{9}-\frac{29\!\cdots\!81}{11\!\cdots\!91}a^{8}-\frac{54\!\cdots\!87}{11\!\cdots\!91}a^{7}+\frac{17\!\cdots\!44}{11\!\cdots\!91}a^{6}-\frac{10\!\cdots\!65}{11\!\cdots\!91}a^{5}+\frac{33\!\cdots\!28}{11\!\cdots\!91}a^{4}-\frac{40\!\cdots\!18}{11\!\cdots\!91}a^{3}+\frac{57\!\cdots\!13}{11\!\cdots\!91}a^{2}-\frac{26\!\cdots\!51}{11\!\cdots\!91}a+\frac{24\!\cdots\!97}{11\!\cdots\!91}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

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

Class group and class number

Trivial group, which has order $1$

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:  $8$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) 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:   $\frac{10\!\cdots\!32}{11\!\cdots\!91}a^{16}-\frac{13\!\cdots\!38}{11\!\cdots\!91}a^{15}+\frac{153497539627694}{11\!\cdots\!91}a^{14}-\frac{20\!\cdots\!47}{11\!\cdots\!91}a^{13}+\frac{73\!\cdots\!74}{11\!\cdots\!91}a^{12}+\frac{19\!\cdots\!19}{11\!\cdots\!91}a^{11}+\frac{70\!\cdots\!54}{11\!\cdots\!91}a^{10}+\frac{19\!\cdots\!26}{11\!\cdots\!91}a^{9}+\frac{66\!\cdots\!91}{11\!\cdots\!91}a^{8}+\frac{15\!\cdots\!03}{11\!\cdots\!91}a^{7}-\frac{16\!\cdots\!82}{11\!\cdots\!91}a^{6}-\frac{10\!\cdots\!83}{11\!\cdots\!91}a^{5}-\frac{35\!\cdots\!48}{11\!\cdots\!91}a^{4}+\frac{42\!\cdots\!32}{11\!\cdots\!91}a^{3}+\frac{77\!\cdots\!97}{11\!\cdots\!91}a^{2}+\frac{35\!\cdots\!26}{11\!\cdots\!91}a-\frac{76\!\cdots\!89}{11\!\cdots\!91}$, $\frac{238227155182776}{11\!\cdots\!91}a^{16}-\frac{820973558914092}{11\!\cdots\!91}a^{15}+\frac{437201946462560}{11\!\cdots\!91}a^{14}+\frac{214894272296728}{11\!\cdots\!91}a^{13}+\frac{19\!\cdots\!89}{11\!\cdots\!91}a^{12}+\frac{16\!\cdots\!49}{11\!\cdots\!91}a^{11}-\frac{10\!\cdots\!58}{11\!\cdots\!91}a^{10}-\frac{68944875781624}{11\!\cdots\!91}a^{9}+\frac{95\!\cdots\!74}{11\!\cdots\!91}a^{8}-\frac{35\!\cdots\!73}{11\!\cdots\!91}a^{7}-\frac{25\!\cdots\!83}{11\!\cdots\!91}a^{6}+\frac{29\!\cdots\!46}{11\!\cdots\!91}a^{5}-\frac{21\!\cdots\!73}{11\!\cdots\!91}a^{4}-\frac{12\!\cdots\!16}{11\!\cdots\!91}a^{3}+\frac{86\!\cdots\!54}{11\!\cdots\!91}a^{2}-\frac{26\!\cdots\!39}{11\!\cdots\!91}a-\frac{19\!\cdots\!55}{11\!\cdots\!91}$, $\frac{361331936520056}{11\!\cdots\!91}a^{16}-\frac{395500407207728}{11\!\cdots\!91}a^{15}-\frac{404488405846192}{11\!\cdots\!91}a^{14}-\frac{477299961985816}{11\!\cdots\!91}a^{13}+\frac{31\!\cdots\!50}{11\!\cdots\!91}a^{12}+\frac{75\!\cdots\!71}{11\!\cdots\!91}a^{11}+\frac{19\!\cdots\!99}{11\!\cdots\!91}a^{10}-\frac{14\!\cdots\!16}{11\!\cdots\!91}a^{9}+\frac{22\!\cdots\!02}{11\!\cdots\!91}a^{8}+\frac{11\!\cdots\!03}{11\!\cdots\!91}a^{7}-\frac{33\!\cdots\!78}{11\!\cdots\!91}a^{6}-\frac{18\!\cdots\!13}{11\!\cdots\!91}a^{5}+\frac{10\!\cdots\!65}{11\!\cdots\!91}a^{4}+\frac{29\!\cdots\!40}{11\!\cdots\!91}a^{3}+\frac{21\!\cdots\!84}{11\!\cdots\!91}a^{2}-\frac{14\!\cdots\!61}{11\!\cdots\!91}a-\frac{10\!\cdots\!97}{11\!\cdots\!91}$, $\frac{541474688268459}{11\!\cdots\!91}a^{16}-\frac{503105829785995}{11\!\cdots\!91}a^{15}-\frac{759647397584390}{11\!\cdots\!91}a^{14}-\frac{700905168343220}{11\!\cdots\!91}a^{13}+\frac{43\!\cdots\!64}{11\!\cdots\!91}a^{12}+\frac{12\!\cdots\!98}{11\!\cdots\!91}a^{11}+\frac{40\!\cdots\!57}{11\!\cdots\!91}a^{10}-\frac{22\!\cdots\!00}{11\!\cdots\!91}a^{9}+\frac{31\!\cdots\!97}{11\!\cdots\!91}a^{8}+\frac{19\!\cdots\!09}{11\!\cdots\!91}a^{7}-\frac{42\!\cdots\!75}{11\!\cdots\!91}a^{6}-\frac{40\!\cdots\!01}{11\!\cdots\!91}a^{5}+\frac{57\!\cdots\!01}{11\!\cdots\!91}a^{4}+\frac{49\!\cdots\!83}{11\!\cdots\!91}a^{3}+\frac{49\!\cdots\!17}{11\!\cdots\!91}a^{2}-\frac{18\!\cdots\!89}{11\!\cdots\!91}a-\frac{37\!\cdots\!43}{11\!\cdots\!91}$, $\frac{49211265331281}{11\!\cdots\!91}a^{16}-\frac{455679016301548}{11\!\cdots\!91}a^{15}+\frac{604764560884272}{11\!\cdots\!91}a^{14}-\frac{220167784432132}{11\!\cdots\!91}a^{13}+\frac{11\!\cdots\!62}{11\!\cdots\!91}a^{12}-\frac{21\!\cdots\!68}{11\!\cdots\!91}a^{11}-\frac{66\!\cdots\!54}{11\!\cdots\!91}a^{10}+\frac{345221016246508}{11\!\cdots\!91}a^{9}-\frac{26\!\cdots\!13}{11\!\cdots\!91}a^{8}-\frac{23\!\cdots\!71}{11\!\cdots\!91}a^{7}-\frac{21\!\cdots\!05}{11\!\cdots\!91}a^{6}+\frac{12\!\cdots\!43}{11\!\cdots\!91}a^{5}+\frac{73\!\cdots\!29}{11\!\cdots\!91}a^{4}+\frac{10\!\cdots\!93}{11\!\cdots\!91}a^{3}-\frac{18\!\cdots\!40}{11\!\cdots\!91}a^{2}-\frac{20\!\cdots\!23}{11\!\cdots\!91}a+\frac{78\!\cdots\!63}{11\!\cdots\!91}$, $\frac{13\!\cdots\!49}{11\!\cdots\!91}a^{16}-\frac{19\!\cdots\!23}{11\!\cdots\!91}a^{15}-\frac{286237517021309}{11\!\cdots\!91}a^{14}-\frac{23\!\cdots\!39}{11\!\cdots\!91}a^{13}+\frac{10\!\cdots\!77}{11\!\cdots\!91}a^{12}+\frac{27\!\cdots\!63}{11\!\cdots\!91}a^{11}+\frac{14\!\cdots\!32}{11\!\cdots\!91}a^{10}+\frac{98\!\cdots\!34}{11\!\cdots\!91}a^{9}+\frac{75\!\cdots\!04}{11\!\cdots\!91}a^{8}+\frac{38\!\cdots\!35}{11\!\cdots\!91}a^{7}-\frac{72\!\cdots\!01}{11\!\cdots\!91}a^{6}-\frac{55\!\cdots\!00}{11\!\cdots\!91}a^{5}-\frac{28\!\cdots\!26}{11\!\cdots\!91}a^{4}+\frac{11\!\cdots\!10}{11\!\cdots\!91}a^{3}+\frac{90\!\cdots\!75}{11\!\cdots\!91}a^{2}-\frac{48\!\cdots\!36}{11\!\cdots\!91}a-\frac{39\!\cdots\!07}{11\!\cdots\!91}$, $\frac{133348952174370}{11\!\cdots\!91}a^{16}-\frac{621949317058588}{11\!\cdots\!91}a^{15}-\frac{12\!\cdots\!06}{11\!\cdots\!91}a^{14}+\frac{21\!\cdots\!88}{11\!\cdots\!91}a^{13}+\frac{14\!\cdots\!21}{11\!\cdots\!91}a^{12}+\frac{34\!\cdots\!20}{11\!\cdots\!91}a^{11}-\frac{22\!\cdots\!06}{11\!\cdots\!91}a^{10}-\frac{34\!\cdots\!84}{11\!\cdots\!91}a^{9}-\frac{10\!\cdots\!65}{11\!\cdots\!91}a^{8}-\frac{59\!\cdots\!82}{11\!\cdots\!91}a^{7}-\frac{12\!\cdots\!55}{11\!\cdots\!91}a^{6}-\frac{28\!\cdots\!18}{11\!\cdots\!91}a^{5}+\frac{35\!\cdots\!94}{11\!\cdots\!91}a^{4}+\frac{54\!\cdots\!04}{11\!\cdots\!91}a^{3}+\frac{23\!\cdots\!64}{11\!\cdots\!91}a^{2}-\frac{13\!\cdots\!15}{11\!\cdots\!91}a-\frac{12\!\cdots\!23}{11\!\cdots\!91}$, $\frac{636029286616479}{11\!\cdots\!91}a^{16}-\frac{20\!\cdots\!81}{11\!\cdots\!91}a^{15}+\frac{19\!\cdots\!57}{11\!\cdots\!91}a^{14}-\frac{466211932185150}{11\!\cdots\!91}a^{13}+\frac{43\!\cdots\!59}{11\!\cdots\!91}a^{12}+\frac{54\!\cdots\!40}{11\!\cdots\!91}a^{11}-\frac{21\!\cdots\!29}{11\!\cdots\!91}a^{10}+\frac{25\!\cdots\!16}{11\!\cdots\!91}a^{9}+\frac{37\!\cdots\!47}{11\!\cdots\!91}a^{8}-\frac{58\!\cdots\!17}{11\!\cdots\!91}a^{7}+\frac{15\!\cdots\!46}{11\!\cdots\!91}a^{6}+\frac{76\!\cdots\!77}{11\!\cdots\!91}a^{5}-\frac{11\!\cdots\!17}{11\!\cdots\!91}a^{4}+\frac{58\!\cdots\!54}{11\!\cdots\!91}a^{3}-\frac{83\!\cdots\!13}{11\!\cdots\!91}a^{2}-\frac{63\!\cdots\!59}{11\!\cdots\!91}a+\frac{13\!\cdots\!79}{11\!\cdots\!91}$ Copy content Toggle raw display
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:  \( 600164.932841 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{8}\cdot 600164.932841 \cdot 1}{2\cdot\sqrt{604462909807314587353088}}\cr\approx \mathstrut & 1.87510129816 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^17 - 2*x^16 + 8*x^13 + 16*x^12 - 16*x^11 + 64*x^9 - 32*x^8 - 80*x^7 + 32*x^6 + 40*x^5 + 80*x^4 + 16*x^3 - 128*x^2 - 2*x + 68)
 
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^17 - 2*x^16 + 8*x^13 + 16*x^12 - 16*x^11 + 64*x^9 - 32*x^8 - 80*x^7 + 32*x^6 + 40*x^5 + 80*x^4 + 16*x^3 - 128*x^2 - 2*x + 68, 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^17 - 2*x^16 + 8*x^13 + 16*x^12 - 16*x^11 + 64*x^9 - 32*x^8 - 80*x^7 + 32*x^6 + 40*x^5 + 80*x^4 + 16*x^3 - 128*x^2 - 2*x + 68);
 
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^17 - 2*x^16 + 8*x^13 + 16*x^12 - 16*x^11 + 64*x^9 - 32*x^8 - 80*x^7 + 32*x^6 + 40*x^5 + 80*x^4 + 16*x^3 - 128*x^2 - 2*x + 68);
 
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

$F_{17}$ (as 17T5):

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 272
The 17 conjugacy class representatives for $F_{17}$
Character table for $F_{17}$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.
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 $16{,}\,{\href{/padicField/3.1.0.1}{1} }$ $16{,}\,{\href{/padicField/5.1.0.1}{1} }$ ${\href{/padicField/7.8.0.1}{8} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }$ $16{,}\,{\href{/padicField/11.1.0.1}{1} }$ $16{,}\,{\href{/padicField/13.1.0.1}{1} }$ ${\href{/padicField/17.4.0.1}{4} }^{4}{,}\,{\href{/padicField/17.1.0.1}{1} }$ $16{,}\,{\href{/padicField/19.1.0.1}{1} }$ ${\href{/padicField/23.8.0.1}{8} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }$ $16{,}\,{\href{/padicField/29.1.0.1}{1} }$ $17$ $16{,}\,{\href{/padicField/37.1.0.1}{1} }$ ${\href{/padicField/41.8.0.1}{8} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }$ $16{,}\,{\href{/padicField/43.1.0.1}{1} }$ ${\href{/padicField/47.4.0.1}{4} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }$ $16{,}\,{\href{/padicField/53.1.0.1}{1} }$ $16{,}\,{\href{/padicField/59.1.0.1}{1} }$

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 $\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.16.79.2$x^{16} + 16 x^{14} + 56 x^{12} + 48 x^{10} + 4 x^{8} + 32 x^{5} + 32 x^{4} + 32 x^{2} + 2$$16$$1$$79$$C_{16}$$[3, 4, 5, 6]$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.8.2t1.a.a$1$ $ 2^{3}$ \(\Q(\sqrt{2}) \) $C_2$ (as 2T1) $1$ $1$
1.16.4t1.a.a$1$ $ 2^{4}$ \(\Q(\zeta_{16})^+\) $C_4$ (as 4T1) $0$ $1$
1.16.4t1.a.b$1$ $ 2^{4}$ \(\Q(\zeta_{16})^+\) $C_4$ (as 4T1) $0$ $1$
1.32.8t1.a.a$1$ $ 2^{5}$ \(\Q(\zeta_{32})^+\) $C_8$ (as 8T1) $0$ $1$
1.32.8t1.a.b$1$ $ 2^{5}$ \(\Q(\zeta_{32})^+\) $C_8$ (as 8T1) $0$ $1$
1.32.8t1.a.c$1$ $ 2^{5}$ \(\Q(\zeta_{32})^+\) $C_8$ (as 8T1) $0$ $1$
1.32.8t1.a.d$1$ $ 2^{5}$ \(\Q(\zeta_{32})^+\) $C_8$ (as 8T1) $0$ $1$
1.64.16t1.a.a$1$ $ 2^{6}$ 16.0.604462909807314587353088.1 $C_{16}$ (as 16T1) $0$ $-1$
1.64.16t1.a.b$1$ $ 2^{6}$ 16.0.604462909807314587353088.1 $C_{16}$ (as 16T1) $0$ $-1$
1.64.16t1.a.c$1$ $ 2^{6}$ 16.0.604462909807314587353088.1 $C_{16}$ (as 16T1) $0$ $-1$
1.64.16t1.a.d$1$ $ 2^{6}$ 16.0.604462909807314587353088.1 $C_{16}$ (as 16T1) $0$ $-1$
1.64.16t1.a.e$1$ $ 2^{6}$ 16.0.604462909807314587353088.1 $C_{16}$ (as 16T1) $0$ $-1$
1.64.16t1.a.f$1$ $ 2^{6}$ 16.0.604462909807314587353088.1 $C_{16}$ (as 16T1) $0$ $-1$
1.64.16t1.a.g$1$ $ 2^{6}$ 16.0.604462909807314587353088.1 $C_{16}$ (as 16T1) $0$ $-1$
1.64.16t1.a.h$1$ $ 2^{6}$ 16.0.604462909807314587353088.1 $C_{16}$ (as 16T1) $0$ $-1$
* 16.604...088.17t5.a.a$16$ $ 2^{79}$ 17.1.604462909807314587353088.1 $F_{17}$ (as 17T5) $1$ $0$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.

Additional information

This field is remarkable in that it is only ramified at 2, and

See Theorem 2.25 of [MR:1299733].