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Properties

Label 32.0.87309122741...0000.7
Degree $32$
Signature $[0, 16]$
Discriminant $2^{128}\cdot 3^{16}\cdot 5^{24}$
Root discriminant $92.66$
Ramified primes $2, 3, 5$
Class number Not computed
Class group Not computed
Galois group $C_4\times C_8$ (as 32T43)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![43046721, 0, 2754990144, 0, 24412273776, 0, 71782798752, 0, 106762245372, 0, 95149196640, 0, 55533931128, 0, 22378468752, 0, 6433893747, 0, 1345652352, 0, 206569440, 0, 23250240, 0, 1895400, 0, 108864, 0, 4176, 0, 96, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 96*x^30 + 4176*x^28 + 108864*x^26 + 1895400*x^24 + 23250240*x^22 + 206569440*x^20 + 1345652352*x^18 + 6433893747*x^16 + 22378468752*x^14 + 55533931128*x^12 + 95149196640*x^10 + 106762245372*x^8 + 71782798752*x^6 + 24412273776*x^4 + 2754990144*x^2 + 43046721)
 
gp: K = bnfinit(x^32 + 96*x^30 + 4176*x^28 + 108864*x^26 + 1895400*x^24 + 23250240*x^22 + 206569440*x^20 + 1345652352*x^18 + 6433893747*x^16 + 22378468752*x^14 + 55533931128*x^12 + 95149196640*x^10 + 106762245372*x^8 + 71782798752*x^6 + 24412273776*x^4 + 2754990144*x^2 + 43046721, 1)
 

Normalized defining polynomial

\( x^{32} + 96 x^{30} + 4176 x^{28} + 108864 x^{26} + 1895400 x^{24} + 23250240 x^{22} + 206569440 x^{20} + 1345652352 x^{18} + 6433893747 x^{16} + 22378468752 x^{14} + 55533931128 x^{12} + 95149196640 x^{10} + 106762245372 x^{8} + 71782798752 x^{6} + 24412273776 x^{4} + 2754990144 x^{2} + 43046721 \)

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:  \(873091227416114037923609044826021953536000000000000000000000000=2^{128}\cdot 3^{16}\cdot 5^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $92.66$
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:  \(480=2^{5}\cdot 3\cdot 5\)
Dirichlet character group:    $\lbrace$$\chi_{480}(1,·)$, $\chi_{480}(389,·)$, $\chi_{480}(7,·)$, $\chi_{480}(269,·)$, $\chi_{480}(149,·)$, $\chi_{480}(409,·)$, $\chi_{480}(29,·)$, $\chi_{480}(289,·)$, $\chi_{480}(169,·)$, $\chi_{480}(49,·)$, $\chi_{480}(443,·)$, $\chi_{480}(323,·)$, $\chi_{480}(203,·)$, $\chi_{480}(461,·)$, $\chi_{480}(463,·)$, $\chi_{480}(83,·)$, $\chi_{480}(341,·)$, $\chi_{480}(343,·)$, $\chi_{480}(347,·)$, $\chi_{480}(221,·)$, $\chi_{480}(223,·)$, $\chi_{480}(227,·)$, $\chi_{480}(101,·)$, $\chi_{480}(103,·)$, $\chi_{480}(361,·)$, $\chi_{480}(107,·)$, $\chi_{480}(367,·)$, $\chi_{480}(241,·)$, $\chi_{480}(467,·)$, $\chi_{480}(247,·)$, $\chi_{480}(121,·)$, $\chi_{480}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{3} a^{2}$, $\frac{1}{3} a^{3}$, $\frac{1}{9} a^{4}$, $\frac{1}{9} a^{5}$, $\frac{1}{27} a^{6}$, $\frac{1}{27} a^{7}$, $\frac{1}{81} a^{8}$, $\frac{1}{81} a^{9}$, $\frac{1}{243} a^{10}$, $\frac{1}{243} a^{11}$, $\frac{1}{729} a^{12}$, $\frac{1}{729} a^{13}$, $\frac{1}{2187} a^{14}$, $\frac{1}{2187} a^{15}$, $\frac{1}{6561} a^{16}$, $\frac{1}{6561} a^{17}$, $\frac{1}{19683} a^{18}$, $\frac{1}{19683} a^{19}$, $\frac{1}{59049} a^{20}$, $\frac{1}{59049} a^{21}$, $\frac{1}{177147} a^{22}$, $\frac{1}{177147} a^{23}$, $\frac{1}{531441} a^{24}$, $\frac{1}{531441} a^{25}$, $\frac{1}{1594323} a^{26}$, $\frac{1}{1594323} a^{27}$, $\frac{1}{4782969} a^{28}$, $\frac{1}{4782969} a^{29}$, $\frac{1}{14348907} a^{30}$, $\frac{1}{14348907} a^{31}$

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

Class group and class number

Not computed

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:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\times C_8$ (as 32T43):

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 $C_4\times C_8$
Character table for $C_4\times C_8$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), \(\Q(\zeta_{20})^+\), \(\Q(\sqrt{2}, \sqrt{5})\), 4.4.8000.1, \(\Q(\zeta_{16})^+\), 4.4.51200.1, 4.4.256000.1, 4.4.256000.2, \(\Q(\zeta_{40})^+\), 8.8.2621440000.1, 8.8.65536000000.1, 8.0.173946175488.1, 8.0.108716359680000.13, 8.0.2717908992000000.4, 8.0.2717908992000000.10, \(\Q(\zeta_{80})^+\), 16.0.11819246862071129702400000000.1, 16.0.7387029288794456064000000000000.3

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{4}$

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$3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
5Data not computed