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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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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content 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)
 
Copy content 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)
 
Copy content 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);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(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

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:  $17$
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:  $(1, 8)$
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:   \(604462909807314587353088\) \(\medspace = 2^{79}\) 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:  \(25.06\)
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:  $2^{79/16}\approx 30.64330498235436$
Ramified primes:   \(2\) 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:  \(\Q(\sqrt{2}) \)
$\Aut(K/\Q)$:   $C_1$
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 \(a\))

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

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

Class number formula

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

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^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))))
 
Copy content gp:\\ 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)))]
 
Copy content magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<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); 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^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):

Copy content comment:Galois group
 
Copy content sage:K.galois_group()
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:G = 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 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 $\Q$.
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]
 

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.

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 $\Q_{2}$$x + 1$$1$$1$$0$Trivial$$[\ ]$$
2.1.16.79a1.38387$x^{16} + 8 x^{12} + 32 x^{11} + 16 x^{10} + 32 x^{9} + 4 x^{8} + 32 x^{7} + 32 x^{5} + 2$$16$$1$$79$$C_{16}$$$[3, 4, 5, 6]$$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
*272 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) $0$ $-1$
*272 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 *.

Spectrum of ring of integers

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

Additional information

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

See Theorem 2.25 of [MR:1299733].