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Properties

Label 32.0.20328239244...0000.4
Degree $32$
Signature $[0, 16]$
Discriminant $2^{96}\cdot 3^{16}\cdot 5^{24}$
Root discriminant $46.33$
Ramified primes $2, 3, 5$
Class number $128$ (GRH)
Class group $[8, 16]$ (GRH)
Galois group $C_2\times C_4^2$ (as 32T36)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 0, 0, 0, 0, 0, 0, 274084, 0, 0, 0, 0, 0, 0, 0, 30126, 0, 0, 0, 0, 0, 0, 0, 349, 0, 0, 0, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 349*x^24 + 30126*x^16 + 274084*x^8 + 1)
 
gp: K = bnfinit(x^32 + 349*x^24 + 30126*x^16 + 274084*x^8 + 1, 1)
 

Normalized defining polynomial

\( x^{32} + 349 x^{24} + 30126 x^{16} + 274084 x^{8} + 1 \)

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

Invariants

Degree:  $32$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 16]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(203282392447840896882957090816000000000000000000000000=2^{96}\cdot 3^{16}\cdot 5^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $46.33$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 3, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(240=2^{4}\cdot 3\cdot 5\)
Dirichlet character group:    $\lbrace$$\chi_{240}(1,·)$, $\chi_{240}(137,·)$, $\chi_{240}(139,·)$, $\chi_{240}(143,·)$, $\chi_{240}(17,·)$, $\chi_{240}(19,·)$, $\chi_{240}(23,·)$, $\chi_{240}(31,·)$, $\chi_{240}(167,·)$, $\chi_{240}(169,·)$, $\chi_{240}(173,·)$, $\chi_{240}(47,·)$, $\chi_{240}(49,·)$, $\chi_{240}(181,·)$, $\chi_{240}(151,·)$, $\chi_{240}(61,·)$, $\chi_{240}(53,·)$, $\chi_{240}(197,·)$, $\chi_{240}(199,·)$, $\chi_{240}(203,·)$, $\chi_{240}(77,·)$, $\chi_{240}(79,·)$, $\chi_{240}(83,·)$, $\chi_{240}(91,·)$, $\chi_{240}(227,·)$, $\chi_{240}(229,·)$, $\chi_{240}(233,·)$, $\chi_{240}(107,·)$, $\chi_{240}(109,·)$, $\chi_{240}(113,·)$, $\chi_{240}(211,·)$, $\chi_{240}(121,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{3} a^{16} - \frac{1}{3} a^{8} + \frac{1}{3}$, $\frac{1}{3} a^{17} - \frac{1}{3} a^{9} + \frac{1}{3} a$, $\frac{1}{3} a^{18} - \frac{1}{3} a^{10} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{19} - \frac{1}{3} a^{11} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{20} - \frac{1}{3} a^{12} + \frac{1}{3} a^{4}$, $\frac{1}{3} a^{21} - \frac{1}{3} a^{13} + \frac{1}{3} a^{5}$, $\frac{1}{3} a^{22} - \frac{1}{3} a^{14} + \frac{1}{3} a^{6}$, $\frac{1}{3} a^{23} - \frac{1}{3} a^{15} + \frac{1}{3} a^{7}$, $\frac{1}{5724006747} a^{24} + \frac{164543023}{5724006747} a^{16} - \frac{672577978}{5724006747} a^{8} - \frac{431059000}{5724006747}$, $\frac{1}{5724006747} a^{25} + \frac{164543023}{5724006747} a^{17} - \frac{672577978}{5724006747} a^{9} - \frac{431059000}{5724006747} a$, $\frac{1}{5724006747} a^{26} + \frac{164543023}{5724006747} a^{18} - \frac{672577978}{5724006747} a^{10} - \frac{431059000}{5724006747} a^{2}$, $\frac{1}{5724006747} a^{27} + \frac{164543023}{5724006747} a^{19} - \frac{672577978}{5724006747} a^{11} - \frac{431059000}{5724006747} a^{3}$, $\frac{1}{5724006747} a^{28} + \frac{164543023}{5724006747} a^{20} - \frac{672577978}{5724006747} a^{12} - \frac{431059000}{5724006747} a^{4}$, $\frac{1}{5724006747} a^{29} + \frac{164543023}{5724006747} a^{21} - \frac{672577978}{5724006747} a^{13} - \frac{431059000}{5724006747} a^{5}$, $\frac{1}{5724006747} a^{30} + \frac{164543023}{5724006747} a^{22} - \frac{672577978}{5724006747} a^{14} - \frac{431059000}{5724006747} a^{6}$, $\frac{1}{5724006747} a^{31} + \frac{164543023}{5724006747} a^{23} - \frac{672577978}{5724006747} a^{15} - \frac{431059000}{5724006747} a^{7}$

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

Class group and class number

$C_{8}\times C_{16}$, which has order $128$ (assuming GRH)

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $15$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{764924}{1908002249} a^{27} - \frac{800905403}{5724006747} a^{19} - \frac{69123602032}{5724006747} a^{11} - \frac{626479158983}{5724006747} a^{3} \) (order $16$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
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)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 1099105261005.4811 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_4^2$ (as 32T36):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 32
The 32 conjugacy class representatives for $C_2\times C_4^2$
Character table for $C_2\times C_4^2$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), \(\Q(\zeta_{8})\), \(\Q(\zeta_{16})^+\), 4.0.2048.2, \(\Q(i, \sqrt{10})\), \(\Q(i, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{-5})\), 4.4.51200.1, 4.0.51200.2, \(\Q(\sqrt{-2}, \sqrt{-5})\), \(\Q(\sqrt{-2}, \sqrt{5})\), 4.4.2304000.2, 4.0.2304000.2, 4.4.2304000.1, 4.0.2304000.1, \(\Q(\zeta_{15})^+\), 4.0.18000.1, 4.4.72000.1, 4.0.72000.2, \(\Q(\zeta_{16})\), 8.0.40960000.1, 8.0.10485760000.3, 8.8.2621440000.1, 8.0.10485760000.2, 8.0.2621440000.1, 8.0.10485760000.1, 8.0.21233664000000.8, 8.0.21233664000000.7, 8.0.324000000.1, 8.0.82944000000.7, 8.8.5308416000000.1, 8.0.5308416000000.7, 8.8.5184000000.1, 8.0.82944000000.6, 8.0.21233664000000.5, 8.0.21233664000000.6, 8.0.5184000000.2, 8.0.82944000000.5, 16.0.109951162777600000000.1, 16.0.450868486864896000000000000.3, 16.0.6879707136000000000000.3, 16.16.28179280429056000000000000.2, 16.0.450868486864896000000000000.13, 16.0.450868486864896000000000000.12, 16.0.28179280429056000000000000.6

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{8}$

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

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];
 
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]])
 

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}$
5Data not computed