Properties

Label 32.0.156...216.1
Degree $32$
Signature $[0, 16]$
Discriminant $1.567\times 10^{53}$
Root discriminant \(45.96\)
Ramified primes $2,3,11$
Class number not computed
Class group not computed
Galois group $C_2^3\times C_4$ (as 32T34)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 113*x^24 + 6208*x^16 + 741393*x^8 + 43046721)
 
gp: K = bnfinit(y^32 + 113*y^24 + 6208*y^16 + 741393*y^8 + 43046721, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 + 113*x^24 + 6208*x^16 + 741393*x^8 + 43046721);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^32 + 113*x^24 + 6208*x^16 + 741393*x^8 + 43046721)
 

\( x^{32} + 113x^{24} + 6208x^{16} + 741393x^{8} + 43046721 \) 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:   \(156712132992285501115475016655592096377279059345801216\) \(\medspace = 2^{96}\cdot 3^{16}\cdot 11^{16}\) 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:  \(45.96\)
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:  $2^{3}3^{1/2}11^{1/2}\approx 45.95650117230423$
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:  \(\Q\)
$\card{ \Gal(K/\Q) }$:  $32$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(528=2^{4}\cdot 3\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{528}(1,·)$, $\chi_{528}(131,·)$, $\chi_{528}(133,·)$, $\chi_{528}(263,·)$, $\chi_{528}(265,·)$, $\chi_{528}(395,·)$, $\chi_{528}(397,·)$, $\chi_{528}(527,·)$, $\chi_{528}(23,·)$, $\chi_{528}(155,·)$, $\chi_{528}(287,·)$, $\chi_{528}(419,·)$, $\chi_{528}(43,·)$, $\chi_{528}(175,·)$, $\chi_{528}(307,·)$, $\chi_{528}(439,·)$, $\chi_{528}(65,·)$, $\chi_{528}(67,·)$, $\chi_{528}(197,·)$, $\chi_{528}(199,·)$, $\chi_{528}(329,·)$, $\chi_{528}(331,·)$, $\chi_{528}(461,·)$, $\chi_{528}(463,·)$, $\chi_{528}(89,·)$, $\chi_{528}(221,·)$, $\chi_{528}(353,·)$, $\chi_{528}(485,·)$, $\chi_{528}(109,·)$, $\chi_{528}(241,·)$, $\chi_{528}(373,·)$, $\chi_{528}(505,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{32768}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{35}a^{16}+\frac{4}{35}a^{8}+\frac{16}{35}$, $\frac{1}{105}a^{17}-\frac{31}{105}a^{9}+\frac{16}{105}a$, $\frac{1}{315}a^{18}-\frac{31}{315}a^{10}+\frac{16}{315}a^{2}$, $\frac{1}{945}a^{19}-\frac{346}{945}a^{11}-\frac{299}{945}a^{3}$, $\frac{1}{2835}a^{20}+\frac{599}{2835}a^{12}-\frac{1244}{2835}a^{4}$, $\frac{1}{8505}a^{21}+\frac{599}{8505}a^{13}+\frac{1591}{8505}a^{5}$, $\frac{1}{25515}a^{22}-\frac{7906}{25515}a^{14}-\frac{6914}{25515}a^{6}$, $\frac{1}{76545}a^{23}+\frac{17609}{76545}a^{15}-\frac{6914}{76545}a^{7}$, $\frac{1}{1425574080}a^{24}+\frac{929}{229635}a^{16}-\frac{98414}{229635}a^{8}+\frac{62193}{217280}$, $\frac{1}{4276722240}a^{25}+\frac{929}{688905}a^{17}-\frac{98414}{688905}a^{9}+\frac{279473}{651840}a$, $\frac{1}{12830166720}a^{26}+\frac{929}{2066715}a^{18}-\frac{98414}{2066715}a^{10}+\frac{931313}{1955520}a^{2}$, $\frac{1}{38490500160}a^{27}+\frac{929}{6200145}a^{19}+\frac{1968301}{6200145}a^{11}+\frac{931313}{5866560}a^{3}$, $\frac{1}{115471500480}a^{28}+\frac{929}{18600435}a^{20}+\frac{8168446}{18600435}a^{12}+\frac{931313}{17599680}a^{4}$, $\frac{1}{346414501440}a^{29}+\frac{929}{55801305}a^{21}-\frac{10431989}{55801305}a^{13}+\frac{931313}{52799040}a^{5}$, $\frac{1}{1039243504320}a^{30}+\frac{929}{167403915}a^{22}-\frac{66233294}{167403915}a^{14}+\frac{931313}{158397120}a^{6}$, $\frac{1}{3117730512960}a^{31}+\frac{929}{502211745}a^{23}-\frac{66233294}{502211745}a^{15}-\frac{157465807}{475191360}a^{7}$ 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:   \( \frac{28589}{38490500160} a^{27} + \frac{253}{6200145} a^{19} + \frac{15467}{6200145} a^{11} + \frac{61479}{217280} a^{3} \)  (order $48$) 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 + 113*x^24 + 6208*x^16 + 741393*x^8 + 43046721)
 
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 + 113*x^24 + 6208*x^16 + 741393*x^8 + 43046721, 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 + 113*x^24 + 6208*x^16 + 741393*x^8 + 43046721);
 
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 + 113*x^24 + 6208*x^16 + 741393*x^8 + 43046721);
 
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

$C_2^3\times C_4$ (as 32T34):

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)
 
An abelian group of order 32
The 32 conjugacy class representatives for $C_2^3\times C_4$
Character table for $C_2^3\times C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{33}) \), \(\Q(\sqrt{-33}) \), \(\Q(\sqrt{22}) \), \(\Q(\sqrt{-22}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{66}) \), \(\Q(\sqrt{-66}) \), \(\Q(\sqrt{11}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), \(\Q(i, \sqrt{33})\), \(\Q(i, \sqrt{22})\), \(\Q(i, \sqrt{6})\), \(\Q(\sqrt{6}, \sqrt{22})\), \(\Q(\sqrt{-6}, \sqrt{-22})\), \(\Q(\sqrt{-6}, \sqrt{22})\), \(\Q(\sqrt{6}, \sqrt{-22})\), \(\Q(\zeta_{8})\), \(\Q(i, \sqrt{66})\), \(\Q(i, \sqrt{11})\), \(\Q(\zeta_{12})\), \(\Q(\sqrt{2}, \sqrt{33})\), \(\Q(\sqrt{-2}, \sqrt{33})\), \(\Q(\sqrt{3}, \sqrt{11})\), \(\Q(\sqrt{-3}, \sqrt{-11})\), \(\Q(\sqrt{2}, \sqrt{-33})\), \(\Q(\sqrt{-2}, \sqrt{-33})\), \(\Q(\sqrt{-3}, \sqrt{11})\), \(\Q(\sqrt{3}, \sqrt{-11})\), \(\Q(\sqrt{2}, \sqrt{11})\), \(\Q(\sqrt{-2}, \sqrt{-11})\), \(\Q(\sqrt{3}, \sqrt{22})\), \(\Q(\sqrt{-3}, \sqrt{22})\), \(\Q(\sqrt{2}, \sqrt{-11})\), \(\Q(\sqrt{-2}, \sqrt{11})\), \(\Q(\sqrt{-3}, \sqrt{-22})\), \(\Q(\sqrt{3}, \sqrt{-22})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\sqrt{-2}, \sqrt{-3})\), \(\Q(\sqrt{6}, \sqrt{11})\), \(\Q(\sqrt{6}, \sqrt{-11})\), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\sqrt{-2}, \sqrt{3})\), \(\Q(\sqrt{-6}, \sqrt{-11})\), \(\Q(\sqrt{-6}, \sqrt{11})\), 4.4.247808.1, 4.0.247808.2, 4.4.18432.1, 4.0.18432.2, \(\Q(\zeta_{16})^+\), 4.0.2048.2, 4.4.2230272.1, 4.0.2230272.1, 8.0.77720518656.1, 8.0.77720518656.8, 8.0.303595776.1, 8.0.959512576.1, 8.0.77720518656.9, \(\Q(\zeta_{24})\), 8.0.77720518656.3, 8.8.77720518656.1, 8.0.4857532416.2, 8.0.4857532416.1, 8.0.77720518656.4, 8.0.77720518656.7, 8.0.77720518656.2, 8.0.77720518656.5, 8.0.77720518656.6, 8.0.245635219456.2, 8.0.1358954496.4, \(\Q(\zeta_{16})\), 8.0.19896452775936.87, 8.8.4974113193984.1, 8.0.4974113193984.1, 8.8.4974113193984.2, 8.0.4974113193984.4, 8.0.19896452775936.40, 8.0.19896452775936.41, 8.0.19896452775936.81, 8.0.19896452775936.79, 8.8.245635219456.1, 8.0.245635219456.1, 8.8.19896452775936.3, 8.0.19896452775936.72, 8.0.61408804864.1, 8.0.61408804864.2, 8.0.4974113193984.3, 8.0.4974113193984.2, 8.8.19896452775936.2, 8.0.19896452775936.77, \(\Q(\zeta_{48})^+\), 8.0.1358954496.3, 8.0.4974113193984.6, 8.0.4974113193984.5, 8.0.339738624.1, 8.0.339738624.2, 16.0.6040479020157644046336.1, 16.0.395868833065051360220676096.5, 16.0.395868833065051360220676096.3, 16.0.60336661037197280935936.1, 16.0.395868833065051360220676096.9, 16.0.395868833065051360220676096.4, \(\Q(\zeta_{48})\), 16.16.395868833065051360220676096.1, 16.0.395868833065051360220676096.1, 16.0.24741802066565710013792256.1, 16.0.24741802066565710013792256.2, 16.0.395868833065051360220676096.8, 16.0.395868833065051360220676096.7, 16.0.395868833065051360220676096.6, 16.0.395868833065051360220676096.2

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 ${\href{/padicField/5.4.0.1}{4} }^{8}$ ${\href{/padicField/7.2.0.1}{2} }^{16}$ R ${\href{/padicField/13.4.0.1}{4} }^{8}$ ${\href{/padicField/17.2.0.1}{2} }^{16}$ ${\href{/padicField/19.4.0.1}{4} }^{8}$ ${\href{/padicField/23.2.0.1}{2} }^{16}$ ${\href{/padicField/29.4.0.1}{4} }^{8}$ ${\href{/padicField/31.2.0.1}{2} }^{16}$ ${\href{/padicField/37.4.0.1}{4} }^{8}$ ${\href{/padicField/41.2.0.1}{2} }^{16}$ ${\href{/padicField/43.4.0.1}{4} }^{8}$ ${\href{/padicField/47.2.0.1}{2} }^{16}$ ${\href{/padicField/53.4.0.1}{4} }^{8}$ ${\href{/padicField/59.4.0.1}{4} }^{8}$

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 2.16.48.1$x^{16} - 8 x^{15} + 64 x^{14} + 8 x^{13} + 76 x^{12} + 48 x^{11} + 64 x^{10} + 256 x^{9} + 56 x^{8} + 144 x^{7} + 160 x^{6} + 432 x^{5} + 456 x^{4} + 256 x^{2} + 288 x + 516$$8$$2$$48$$C_4\times C_2^2$$[2, 3, 4]^{2}$
2.16.48.1$x^{16} - 8 x^{15} + 64 x^{14} + 8 x^{13} + 76 x^{12} + 48 x^{11} + 64 x^{10} + 256 x^{9} + 56 x^{8} + 144 x^{7} + 160 x^{6} + 432 x^{5} + 456 x^{4} + 256 x^{2} + 288 x + 516$$8$$2$$48$$C_4\times C_2^2$$[2, 3, 4]^{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}$
\(11\) Copy content Toggle raw display 11.8.4.1$x^{8} + 60 x^{6} + 20 x^{5} + 970 x^{4} - 280 x^{3} + 4664 x^{2} - 5460 x + 2325$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
11.8.4.1$x^{8} + 60 x^{6} + 20 x^{5} + 970 x^{4} - 280 x^{3} + 4664 x^{2} - 5460 x + 2325$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
11.8.4.1$x^{8} + 60 x^{6} + 20 x^{5} + 970 x^{4} - 280 x^{3} + 4664 x^{2} - 5460 x + 2325$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
11.8.4.1$x^{8} + 60 x^{6} + 20 x^{5} + 970 x^{4} - 280 x^{3} + 4664 x^{2} - 5460 x + 2325$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$