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Properties

Label 32.0.620...288.1
Degree $32$
Signature $[0, 16]$
Discriminant $6.208\times 10^{81}$
Root discriminant \(359.77\)
Ramified primes $2,3,11$
Class number not computed
Class group not computed
Galois group $C_{32}$ (as 32T33)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 1056*x^30 + 505296*x^28 + 144897984*x^26 + 27750551400*x^24 + 3744474402240*x^22 + 365950363695840*x^20 + 26222957489976192*x^18 + 1379163670488435348*x^16 + 52768001305644482880*x^14 + 1440566435644094382624*x^12 + 27164967072145779786624*x^10 + 336166467517804024859472*x^8 + 2515124016084622825669248*x^6 + 9880844348903875386557760*x^4 + 15344370047709547659124992*x^2 + 3955970402925117755868162)
 
gp: K = bnfinit(y^32 + 1056*y^30 + 505296*y^28 + 144897984*y^26 + 27750551400*y^24 + 3744474402240*y^22 + 365950363695840*y^20 + 26222957489976192*y^18 + 1379163670488435348*y^16 + 52768001305644482880*y^14 + 1440566435644094382624*y^12 + 27164967072145779786624*y^10 + 336166467517804024859472*y^8 + 2515124016084622825669248*y^6 + 9880844348903875386557760*y^4 + 15344370047709547659124992*y^2 + 3955970402925117755868162, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 + 1056*x^30 + 505296*x^28 + 144897984*x^26 + 27750551400*x^24 + 3744474402240*x^22 + 365950363695840*x^20 + 26222957489976192*x^18 + 1379163670488435348*x^16 + 52768001305644482880*x^14 + 1440566435644094382624*x^12 + 27164967072145779786624*x^10 + 336166467517804024859472*x^8 + 2515124016084622825669248*x^6 + 9880844348903875386557760*x^4 + 15344370047709547659124992*x^2 + 3955970402925117755868162);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^32 + 1056*x^30 + 505296*x^28 + 144897984*x^26 + 27750551400*x^24 + 3744474402240*x^22 + 365950363695840*x^20 + 26222957489976192*x^18 + 1379163670488435348*x^16 + 52768001305644482880*x^14 + 1440566435644094382624*x^12 + 27164967072145779786624*x^10 + 336166467517804024859472*x^8 + 2515124016084622825669248*x^6 + 9880844348903875386557760*x^4 + 15344370047709547659124992*x^2 + 3955970402925117755868162)
 

\( x^{32} + 1056 x^{30} + 505296 x^{28} + 144897984 x^{26} + 27750551400 x^{24} + 3744474402240 x^{22} + \cdots + 39\!\cdots\!62 \) 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:   \(620\!\cdots\!288\) \(\medspace = 2^{191}\cdot 3^{16}\cdot 11^{16}\) 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:  \(359.77\)
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:  $2^{191/32}3^{1/2}11^{1/2}\approx 359.7739849481405$
Ramified primes:   \(2\), \(3\), \(11\) 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:  \(\Q(\sqrt{2}) \)
$\card{ \Gal(K/\Q) }$:  $32$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4224=2^{7}\cdot 3\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{4224}(1,·)$, $\chi_{4224}(131,·)$, $\chi_{4224}(265,·)$, $\chi_{4224}(395,·)$, $\chi_{4224}(529,·)$, $\chi_{4224}(659,·)$, $\chi_{4224}(793,·)$, $\chi_{4224}(923,·)$, $\chi_{4224}(1057,·)$, $\chi_{4224}(1187,·)$, $\chi_{4224}(1321,·)$, $\chi_{4224}(1451,·)$, $\chi_{4224}(1585,·)$, $\chi_{4224}(1715,·)$, $\chi_{4224}(1849,·)$, $\chi_{4224}(1979,·)$, $\chi_{4224}(2113,·)$, $\chi_{4224}(2243,·)$, $\chi_{4224}(2377,·)$, $\chi_{4224}(2507,·)$, $\chi_{4224}(2641,·)$, $\chi_{4224}(2771,·)$, $\chi_{4224}(2905,·)$, $\chi_{4224}(3035,·)$, $\chi_{4224}(3169,·)$, $\chi_{4224}(3299,·)$, $\chi_{4224}(3433,·)$, $\chi_{4224}(3563,·)$, $\chi_{4224}(3697,·)$, $\chi_{4224}(3827,·)$, $\chi_{4224}(3961,·)$, $\chi_{4224}(4091,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{32768}$

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

$1$, $a$, $\frac{1}{33}a^{2}$, $\frac{1}{33}a^{3}$, $\frac{1}{1089}a^{4}$, $\frac{1}{1089}a^{5}$, $\frac{1}{35937}a^{6}$, $\frac{1}{35937}a^{7}$, $\frac{1}{1185921}a^{8}$, $\frac{1}{1185921}a^{9}$, $\frac{1}{39135393}a^{10}$, $\frac{1}{39135393}a^{11}$, $\frac{1}{1291467969}a^{12}$, $\frac{1}{1291467969}a^{13}$, $\frac{1}{42618442977}a^{14}$, $\frac{1}{42618442977}a^{15}$, $\frac{1}{1406408618241}a^{16}$, $\frac{1}{1406408618241}a^{17}$, $\frac{1}{46411484401953}a^{18}$, $\frac{1}{46411484401953}a^{19}$, $\frac{1}{15\!\cdots\!49}a^{20}$, $\frac{1}{15\!\cdots\!49}a^{21}$, $\frac{1}{50\!\cdots\!17}a^{22}$, $\frac{1}{50\!\cdots\!17}a^{23}$, $\frac{1}{16\!\cdots\!61}a^{24}$, $\frac{1}{16\!\cdots\!61}a^{25}$, $\frac{1}{55\!\cdots\!13}a^{26}$, $\frac{1}{55\!\cdots\!13}a^{27}$, $\frac{1}{18\!\cdots\!29}a^{28}$, $\frac{1}{18\!\cdots\!29}a^{29}$, $\frac{1}{59\!\cdots\!57}a^{30}$, $\frac{1}{59\!\cdots\!57}a^{31}$ 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

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:   \( -1 \)  (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:  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 + 1056*x^30 + 505296*x^28 + 144897984*x^26 + 27750551400*x^24 + 3744474402240*x^22 + 365950363695840*x^20 + 26222957489976192*x^18 + 1379163670488435348*x^16 + 52768001305644482880*x^14 + 1440566435644094382624*x^12 + 27164967072145779786624*x^10 + 336166467517804024859472*x^8 + 2515124016084622825669248*x^6 + 9880844348903875386557760*x^4 + 15344370047709547659124992*x^2 + 3955970402925117755868162)
 
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 + 1056*x^30 + 505296*x^28 + 144897984*x^26 + 27750551400*x^24 + 3744474402240*x^22 + 365950363695840*x^20 + 26222957489976192*x^18 + 1379163670488435348*x^16 + 52768001305644482880*x^14 + 1440566435644094382624*x^12 + 27164967072145779786624*x^10 + 336166467517804024859472*x^8 + 2515124016084622825669248*x^6 + 9880844348903875386557760*x^4 + 15344370047709547659124992*x^2 + 3955970402925117755868162, 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 + 1056*x^30 + 505296*x^28 + 144897984*x^26 + 27750551400*x^24 + 3744474402240*x^22 + 365950363695840*x^20 + 26222957489976192*x^18 + 1379163670488435348*x^16 + 52768001305644482880*x^14 + 1440566435644094382624*x^12 + 27164967072145779786624*x^10 + 336166467517804024859472*x^8 + 2515124016084622825669248*x^6 + 9880844348903875386557760*x^4 + 15344370047709547659124992*x^2 + 3955970402925117755868162);
 
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 + 1056*x^30 + 505296*x^28 + 144897984*x^26 + 27750551400*x^24 + 3744474402240*x^22 + 365950363695840*x^20 + 26222957489976192*x^18 + 1379163670488435348*x^16 + 52768001305644482880*x^14 + 1440566435644094382624*x^12 + 27164967072145779786624*x^10 + 336166467517804024859472*x^8 + 2515124016084622825669248*x^6 + 9880844348903875386557760*x^4 + 15344370047709547659124992*x^2 + 3955970402925117755868162);
 
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

$C_{32}$ (as 32T33):

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 cyclic group of order 32
The 32 conjugacy class representatives for $C_{32}$
Character table for $C_{32}$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\), \(\Q(\zeta_{32})^+\), \(\Q(\zeta_{64})^+\)

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 $32$ $16^{2}$ R $32$ ${\href{/padicField/17.8.0.1}{8} }^{4}$ $32$ $16^{2}$ $32$ ${\href{/padicField/31.4.0.1}{4} }^{8}$ $32$ $16^{2}$ $32$ ${\href{/padicField/47.8.0.1}{8} }^{4}$ $32$ $32$

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 $32$$32$$1$$191$
\(3\) Copy content Toggle raw display Deg $32$$2$$16$$16$
\(11\) Copy content Toggle raw display Deg $32$$2$$16$$16$