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Properties

Label 32.0.48679729995...5376.2
Degree $32$
Signature $[0, 16]$
Discriminant $2^{128}\cdot 3^{16}\cdot 7^{16}$
Root discriminant $73.32$
Ramified primes $2, 3, 7$
Class number $19584$ (GRH)
Class group $[24, 816]$ (GRH)
Galois group $C_2^2\times C_8$ (as 32T37)

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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, 0, 0, 0, 0, 0, 0, 0, 0, 277727, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 277727*x^16 + 1)
 
gp: K = bnfinit(x^32 + 277727*x^16 + 1, 1)
 

Normalized defining polynomial

\( x^{32} + 277727 x^{16} + 1 \)

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:  \(486797299958529611528622959063550881784243829894751113445376=2^{128}\cdot 3^{16}\cdot 7^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $73.32$
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:  $2, 3, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(672=2^{5}\cdot 3\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{672}(1,·)$, $\chi_{672}(643,·)$, $\chi_{672}(517,·)$, $\chi_{672}(139,·)$, $\chi_{672}(13,·)$, $\chi_{672}(167,·)$, $\chi_{672}(659,·)$, $\chi_{672}(533,·)$, $\chi_{672}(155,·)$, $\chi_{672}(29,·)$, $\chi_{672}(671,·)$, $\chi_{672}(545,·)$, $\chi_{672}(295,·)$, $\chi_{672}(41,·)$, $\chi_{672}(307,·)$, $\chi_{672}(181,·)$, $\chi_{672}(323,·)$, $\chi_{672}(197,·)$, $\chi_{672}(503,·)$, $\chi_{672}(463,·)$, $\chi_{672}(337,·)$, $\chi_{672}(335,·)$, $\chi_{672}(377,·)$, $\chi_{672}(169,·)$, $\chi_{672}(475,·)$, $\chi_{672}(349,·)$, $\chi_{672}(209,·)$, $\chi_{672}(491,·)$, $\chi_{672}(365,·)$, $\chi_{672}(631,·)$, $\chi_{672}(505,·)$, $\chi_{672}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{60605} a^{16} - \frac{12649}{60605}$, $\frac{1}{60605} a^{17} - \frac{12649}{60605} a$, $\frac{1}{60605} a^{18} - \frac{12649}{60605} a^{2}$, $\frac{1}{60605} a^{19} - \frac{12649}{60605} a^{3}$, $\frac{1}{60605} a^{20} - \frac{12649}{60605} a^{4}$, $\frac{1}{60605} a^{21} - \frac{12649}{60605} a^{5}$, $\frac{1}{60605} a^{22} - \frac{12649}{60605} a^{6}$, $\frac{1}{60605} a^{23} - \frac{12649}{60605} a^{7}$, $\frac{1}{60605} a^{24} - \frac{12649}{60605} a^{8}$, $\frac{1}{60605} a^{25} - \frac{12649}{60605} a^{9}$, $\frac{1}{60605} a^{26} - \frac{12649}{60605} a^{10}$, $\frac{1}{60605} a^{27} - \frac{12649}{60605} a^{11}$, $\frac{1}{60605} a^{28} - \frac{12649}{60605} a^{12}$, $\frac{1}{60605} a^{29} - \frac{12649}{60605} a^{13}$, $\frac{1}{60605} a^{30} - \frac{12649}{60605} a^{14}$, $\frac{1}{60605} a^{31} - \frac{12649}{60605} a^{15}$

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

Class group and class number

$C_{24}\times C_{816}$, which has order $19584$ (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:  \( \frac{12649}{60605} a^{30} + \frac{3512968824}{60605} a^{14} \) (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:  \( 94813681955904.33 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_8$ (as 32T37):

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 $C_2^2\times C_8$
Character table for $C_2^2\times C_8$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-21}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-42}) \), \(\Q(\sqrt{42}) \), \(\Q(i, \sqrt{21})\), \(\Q(\zeta_{8})\), \(\Q(i, \sqrt{42})\), \(\Q(\sqrt{2}, \sqrt{-21})\), \(\Q(\sqrt{-2}, \sqrt{-21})\), \(\Q(\sqrt{2}, \sqrt{21})\), \(\Q(\sqrt{-2}, \sqrt{21})\), \(\Q(\zeta_{16})^+\), 4.0.2048.2, 4.0.903168.5, 4.4.903168.2, 8.0.12745506816.8, \(\Q(\zeta_{16})\), 8.0.3262849744896.7, 8.0.3262849744896.5, 8.0.3262849744896.4, 8.8.815712436224.1, 8.0.815712436224.4, 8.8.5156108238848.1, 8.0.5156108238848.1, 8.0.173946175488.1, 8.8.173946175488.1, 16.0.10646188457767892278050816.4, 16.0.106341808682864896865468416.2, 16.0.121029087867608368152576.2, 16.0.697708606768276588334338277376.2, 16.0.697708606768276588334338277376.3, 16.16.174427151692069147083584569344.2, 16.0.174427151692069147083584569344.5

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 ${\href{/LocalNumberField/5.8.0.1}{8} }^{4}$ R ${\href{/LocalNumberField/11.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{32}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{4}$

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
3Data not computed
$7$7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$