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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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(x^32 + x^28 - x^20 - x^16 - x^12 + x^4 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1]);
 

\( x^{32} + x^{28} - x^{20} - x^{16} - x^{12} + x^{4} + 1 \)

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

Invariants

Degree:  $32$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 16]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(47330370277129322496000000000000000000000000\)\(\medspace = 2^{64}\cdot 3^{16}\cdot 5^{24}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $23.17$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $2, 3, 5$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $32$
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.

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

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

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

Class group and class number

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

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $15$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( a \) (order $120$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 5926511257.21094 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{16}\cdot 5926511257.21094 \cdot 4}{120\sqrt{47330370277129322496000000000000000000000000}}\approx 0.169428240847859$ (assuming GRH)

Galois group

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

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
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.

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

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 
magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data not computed
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$