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Properties

Label 32.0.473...000.1
Degree $32$
Signature $[0, 16]$
Discriminant $4.733\times 10^{43}$
Root discriminant \(23.17\)
Ramified primes $2,3,5$
Class number $4$ (GRH)
Class group [4] (GRH)
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 + x^28 - x^20 - x^16 - x^12 + x^4 + 1)
 
gp: K = bnfinit(y^32 + y^28 - y^20 - y^16 - y^12 + y^4 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^32 + x^28 - x^20 - x^16 - x^12 + x^4 + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^32 + x^28 - x^20 - x^16 - x^12 + x^4 + 1)
 

\( x^{32} + x^{28} - x^{20} - x^{16} - x^{12} + x^{4} + 1 \) 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:   \(47330370277129322496000000000000000000000000\) \(\medspace = 2^{64}\cdot 3^{16}\cdot 5^{24}\) 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:  \(23.17\)
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:  $2^{2}3^{1/2}5^{3/4}\approx 23.165843705765383$
Ramified primes:   \(2\), \(3\), \(5\) 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:  \(120=2^{3}\cdot 3\cdot 5\)
Dirichlet character group:    $\lbrace$$\chi_{120}(1,·)$, $\chi_{120}(7,·)$, $\chi_{120}(11,·)$, $\chi_{120}(13,·)$, $\chi_{120}(17,·)$, $\chi_{120}(19,·)$, $\chi_{120}(23,·)$, $\chi_{120}(29,·)$, $\chi_{120}(31,·)$, $\chi_{120}(37,·)$, $\chi_{120}(41,·)$, $\chi_{120}(43,·)$, $\chi_{120}(47,·)$, $\chi_{120}(49,·)$, $\chi_{120}(53,·)$, $\chi_{120}(59,·)$, $\chi_{120}(61,·)$, $\chi_{120}(67,·)$, $\chi_{120}(71,·)$, $\chi_{120}(73,·)$, $\chi_{120}(77,·)$, $\chi_{120}(79,·)$, $\chi_{120}(83,·)$, $\chi_{120}(89,·)$, $\chi_{120}(91,·)$, $\chi_{120}(97,·)$, $\chi_{120}(101,·)$, $\chi_{120}(103,·)$, $\chi_{120}(107,·)$, $\chi_{120}(109,·)$, $\chi_{120}(113,·)$, $\chi_{120}(119,·)$$\rbrace$
This is a CM field.
Reflex fields:  unavailable$^{32768}$

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$ Copy content Toggle raw display

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

Monogenic:  Yes
Index:  $1$
Inessential primes:  None

Class group and class number

$C_{4}$, which has order $4$ (assuming GRH)

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:   \( a \)  (order $120$) 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:   $a^{24}+1$, $a^{16}-1$, $a^{29}-a^{25}-a^{21}-a^{17}+a^{5}+a$, $a^{18}-1$, $a^{6}-1$, $a^{30}-a^{24}-a^{14}+a^{12}-a^{10}+a^{2}$, $a^{31}-a^{27}-a^{25}+a^{9}+a^{7}$, $a^{9}-1$, $a^{9}-a^{6}$, $a^{21}-1$, $a^{29}-a^{28}-a^{24}+a^{16}+a^{12}+a^{8}-1$, $a^{13}-1$, $a^{11}-1$, $a^{31}+a^{30}-a^{10}$, $a^{25}-1$ Copy content Toggle raw display (assuming GRH)
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:  \( 5926511257.21094 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{16}\cdot 5926511257.21094 \cdot 4}{120\cdot\sqrt{47330370277129322496000000000000000000000000}}\cr\approx \mathstrut & 0.169428240847859 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^32 + x^28 - x^20 - x^16 - x^12 + x^4 + 1)
 
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 + x^28 - x^20 - x^16 - x^12 + x^4 + 1, 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 + x^28 - x^20 - x^16 - x^12 + x^4 + 1);
 
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 + x^28 - x^20 - x^16 - x^12 + x^4 + 1);
 
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

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{-30}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{30}) \), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\zeta_{12})\), \(\Q(\sqrt{-2}, \sqrt{-3})\), \(\Q(\zeta_{8})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(i, \sqrt{6})\), \(\Q(\sqrt{-2}, \sqrt{3})\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\sqrt{-3}, \sqrt{10})\), \(\Q(\sqrt{-3}, \sqrt{-5})\), \(\Q(\sqrt{-3}, \sqrt{-10})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{-15})\), \(\Q(\sqrt{2}, \sqrt{-5})\), \(\Q(\sqrt{2}, \sqrt{15})\), \(\Q(\sqrt{5}, \sqrt{-6})\), \(\Q(\sqrt{-6}, \sqrt{10})\), \(\Q(\sqrt{-5}, \sqrt{-6})\), \(\Q(\sqrt{-6}, \sqrt{-10})\), \(\Q(i, \sqrt{5})\), \(\Q(i, \sqrt{15})\), \(\Q(i, \sqrt{10})\), \(\Q(i, \sqrt{30})\), \(\Q(\sqrt{3}, \sqrt{5})\), \(\Q(\sqrt{3}, \sqrt{-5})\), \(\Q(\sqrt{3}, \sqrt{10})\), \(\Q(\sqrt{3}, \sqrt{-10})\), \(\Q(\sqrt{-2}, \sqrt{5})\), \(\Q(\sqrt{-2}, \sqrt{-15})\), \(\Q(\sqrt{-2}, \sqrt{-5})\), \(\Q(\sqrt{-2}, \sqrt{15})\), \(\Q(\sqrt{5}, \sqrt{6})\), \(\Q(\sqrt{6}, \sqrt{-10})\), \(\Q(\sqrt{6}, \sqrt{10})\), \(\Q(\sqrt{-5}, \sqrt{6})\), \(\Q(\zeta_{5})\), \(\Q(\zeta_{15})^+\), 4.0.8000.2, 4.4.72000.1, \(\Q(\zeta_{20})^+\), 4.0.18000.1, 4.4.8000.1, 4.0.72000.2, \(\Q(\zeta_{24})\), 8.0.207360000.1, 8.0.3317760000.3, 8.0.12960000.1, 8.0.3317760000.2, 8.0.207360000.2, 8.0.3317760000.8, 8.0.40960000.1, 8.0.3317760000.4, 8.8.3317760000.1, 8.0.3317760000.6, 8.0.3317760000.9, 8.0.3317760000.5, 8.0.3317760000.7, 8.0.3317760000.1, \(\Q(\zeta_{15})\), 8.0.5184000000.3, 8.0.324000000.2, 8.0.5184000000.4, 8.0.64000000.2, 8.8.5184000000.1, \(\Q(\zeta_{40})^+\), 8.0.82944000000.6, 8.0.5184000000.5, 8.0.5184000000.1, 8.0.82944000000.2, 8.0.82944000000.3, \(\Q(\zeta_{20})\), 8.0.324000000.1, 8.0.1024000000.2, 8.0.82944000000.7, 8.0.324000000.3, \(\Q(\zeta_{60})^+\), 8.0.82944000000.1, 8.8.82944000000.1, 8.0.64000000.1, 8.0.5184000000.2, 8.0.1024000000.1, 8.0.82944000000.5, 8.0.5184000000.6, 8.8.5184000000.2, 8.0.82944000000.4, 8.8.82944000000.2, 16.0.11007531417600000000.1, 16.0.26873856000000000000.2, 16.0.6879707136000000000000.5, \(\Q(\zeta_{60})\), 16.0.6879707136000000000000.6, 16.0.26873856000000000000.1, 16.0.6879707136000000000000.8, \(\Q(\zeta_{40})\), 16.0.6879707136000000000000.3, 16.0.6879707136000000000000.4, \(\Q(\zeta_{120})^+\), 16.0.6879707136000000000000.9, 16.0.6879707136000000000000.2, 16.0.6879707136000000000000.7, 16.0.6879707136000000000000.1

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]
 

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 R ${\href{/padicField/7.4.0.1}{4} }^{8}$ ${\href{/padicField/11.2.0.1}{2} }^{16}$ ${\href{/padicField/13.4.0.1}{4} }^{8}$ ${\href{/padicField/17.4.0.1}{4} }^{8}$ ${\href{/padicField/19.2.0.1}{2} }^{16}$ ${\href{/padicField/23.4.0.1}{4} }^{8}$ ${\href{/padicField/29.2.0.1}{2} }^{16}$ ${\href{/padicField/31.2.0.1}{2} }^{16}$ ${\href{/padicField/37.4.0.1}{4} }^{8}$ ${\href{/padicField/41.2.0.1}{2} }^{16}$ ${\href{/padicField/43.4.0.1}{4} }^{8}$ ${\href{/padicField/47.4.0.1}{4} }^{8}$ ${\href{/padicField/53.4.0.1}{4} }^{8}$ ${\href{/padicField/59.2.0.1}{2} }^{16}$

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 Deg $16$$4$$4$$32$
Deg $16$$4$$4$$32$
\(3\) 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}$
\(5\) Copy content Toggle raw display 5.8.6.1$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} + 16 x^{7} + 104 x^{6} + 352 x^{5} + 674 x^{4} + 784 x^{3} + 776 x^{2} + 928 x + 721$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$