LMFDB - Number field 32.0.113341328678310292663197722067951895712000278265856.1

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 31*x^24 + 705*x^16 + 7936*x^8 + 65536)

gp: K = bnfinit(x^32 + 31*x^24 + 705*x^16 + 7936*x^8 + 65536, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![65536, 0, 0, 0, 0, 0, 0, 0, 7936, 0, 0, 0, 0, 0, 0, 0, 705, 0, 0, 0, 0, 0, 0, 0, 31, 0, 0, 0, 0, 0, 0, 0, 1]);

$x^{32} + 31 x^{24} + 705 x^{16} + 7936 x^{8} + 65536$

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:	$113341328678310292663197722067951895712000278265856$$\medspace = 2^{96}\cdot 3^{16}\cdot 7^{16}$	sage: K.disc() gp: K.disc magma: Discriminant(Integers(K));
Root discriminant:	$36.66$	sage: (K.disc().abs())^(1./K.degree()) gp: abs(K.disc)^(1/poldegree(K.pol)) magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
Ramified primes:	$2, 3, 7$	sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(Integers(K)));
$\|\Gal(K/\Q)\|$:	$32$
This field is Galois and abelian over $\Q$.
Conductor:	$336=2^{4}\cdot 3\cdot 7$
Dirichlet character group:	$\lbrace$$\chi_{336}(1,·)$, $\chi_{336}(265,·)$, $\chi_{336}(139,·)$, $\chi_{336}(13,·)$, $\chi_{336}(281,·)$, $\chi_{336}(155,·)$, $\chi_{336}(29,·)$, $\chi_{336}(293,·)$, $\chi_{336}(167,·)$, $\chi_{336}(41,·)$, $\chi_{336}(43,·)$, $\chi_{336}(125,·)$, $\chi_{336}(307,·)$, $\chi_{336}(181,·)$, $\chi_{336}(55,·)$, $\chi_{336}(323,·)$, $\chi_{336}(197,·)$, $\chi_{336}(71,·)$, $\chi_{336}(335,·)$, $\chi_{336}(209,·)$, $\chi_{336}(83,·)$, $\chi_{336}(85,·)$, $\chi_{336}(223,·)$, $\chi_{336}(97,·)$, $\chi_{336}(295,·)$, $\chi_{336}(239,·)$, $\chi_{336}(113,·)$, $\chi_{336}(211,·)$, $\chi_{336}(169,·)$, $\chi_{336}(251,·)$, $\chi_{336}(253,·)$, $\chi_{336}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{3} a^{16} - \frac{1}{3} a^{8} + \frac{1}{3}$, $\frac{1}{6} a^{17} - \frac{1}{6} a^{9} + \frac{1}{6} a$, $\frac{1}{12} a^{18} - \frac{1}{12} a^{10} + \frac{1}{12} a^{2}$, $\frac{1}{24} a^{19} - \frac{1}{24} a^{11} + \frac{1}{24} a^{3}$, $\frac{1}{48} a^{20} - \frac{1}{48} a^{12} + \frac{1}{48} a^{4}$, $\frac{1}{96} a^{21} - \frac{1}{96} a^{13} + \frac{1}{96} a^{5}$, $\frac{1}{192} a^{22} + \frac{95}{192} a^{14} + \frac{1}{192} a^{6}$, $\frac{1}{384} a^{23} - \frac{97}{384} a^{15} - \frac{191}{384} a^{7}$, $\frac{1}{541440} a^{24} - \frac{11}{256} a^{16} - \frac{85}{256} a^{8} + \frac{736}{2115}$, $\frac{1}{1082880} a^{25} - \frac{11}{512} a^{17} - \frac{85}{512} a^{9} + \frac{368}{2115} a$, $\frac{1}{2165760} a^{26} - \frac{11}{1024} a^{18} - \frac{85}{1024} a^{10} + \frac{184}{2115} a^{2}$, $\frac{1}{4331520} a^{27} - \frac{11}{2048} a^{19} + \frac{939}{2048} a^{11} - \frac{1931}{4230} a^{3}$, $\frac{1}{8663040} a^{28} - \frac{11}{4096} a^{20} + \frac{939}{4096} a^{12} - \frac{1931}{8460} a^{4}$, $\frac{1}{17326080} a^{29} - \frac{11}{8192} a^{21} - \frac{3157}{8192} a^{13} + \frac{6529}{16920} a^{5}$, $\frac{1}{34652160} a^{30} - \frac{11}{16384} a^{22} + \frac{5035}{16384} a^{14} + \frac{6529}{33840} a^{6}$, $\frac{1}{69304320} a^{31} - \frac{11}{32768} a^{23} + \frac{5035}{32768} a^{15} + \frac{6529}{67680} a^{7}$

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

Class group and class number

$C_{4}\times C_{4}\times C_{4}$, which has order $64$ (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:	$ -\frac{7}{270720} a^{31} + \frac{7193}{270720} a^{7} $ (order $48$)	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:	$ 374180447387.275 $ (assuming GRH)	sage: K.regulator() gp: K.reg magma: Regulator(K);

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{16}\cdot 374180447387.275 \cdot 64}{48\sqrt{113341328678310292663197722067951895712000278265856}}\approx 0.276505370236942$ (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

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	${\href{/LocalNumberField/5.4.0.1}{4} }^{8}$	R	${\href{/LocalNumberField/11.4.0.1}{4} }^{8}$	${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$	${\href{/LocalNumberField/17.2.0.1}{2} }^{16}$	${\href{/LocalNumberField/19.4.0.1}{4} }^{8}$	${\href{/LocalNumberField/23.2.0.1}{2} }^{16}$	${\href{/LocalNumberField/29.4.0.1}{4} }^{8}$	${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$	${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$	${\href{/LocalNumberField/41.2.0.1}{2} }^{16}$	${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$	${\href{/LocalNumberField/47.2.0.1}{2} }^{16}$	${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$	${\href{/LocalNumberField/59.4.0.1}{4} }^{8}$

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

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

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
2	Data 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	$x^{8} + 36 x^{4} - 27 x^{2} + 324$	$2$	$4$	$4$	$C_4\times C_2$	$[\ ]_{2}^{4}$
$7$	7.4.2.1	$x^{4} + 35 x^{2} + 441$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	7.4.2.1	$x^{4} + 35 x^{2} + 441$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	7.4.2.1	$x^{4} + 35 x^{2} + 441$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	7.4.2.1	$x^{4} + 35 x^{2} + 441$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	7.4.2.1	$x^{4} + 35 x^{2} + 441$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	7.4.2.1	$x^{4} + 35 x^{2} + 441$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	7.4.2.1	$x^{4} + 35 x^{2} + 441$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	7.4.2.1	$x^{4} + 35 x^{2} + 441$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$

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