Properties

Label 32.0.113...856.1
Degree $32$
Signature $[0, 16]$
Discriminant $1.133\times 10^{50}$
Root discriminant $36.66$
Ramified primes $2, 3, 7$
Class number $64$ (GRH)
Class group $[4, 4, 4]$ (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 + 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]);
 

\(x^{32} + 31 x^{24} + 705 x^{16} + 7936 x^{8} + 65536\)  Toggle raw display

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 \(a\))

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

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 $48$)  Toggle raw display
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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{-21}) \), \(\Q(\sqrt{-14}) \), \(\Q(\sqrt{14}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{42}) \), \(\Q(\sqrt{-42}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{3}) \), \(\Q(i, \sqrt{21})\), \(\Q(i, \sqrt{14})\), \(\Q(i, \sqrt{6})\), \(\Q(\sqrt{-6}, \sqrt{-14})\), \(\Q(\sqrt{6}, \sqrt{14})\), \(\Q(\sqrt{6}, \sqrt{-14})\), \(\Q(\sqrt{-6}, \sqrt{14})\), \(\Q(\zeta_{8})\), \(\Q(i, \sqrt{42})\), \(\Q(i, \sqrt{7})\), \(\Q(\zeta_{12})\), \(\Q(\sqrt{2}, \sqrt{21})\), \(\Q(\sqrt{-2}, \sqrt{21})\), \(\Q(\sqrt{-3}, \sqrt{-7})\), \(\Q(\sqrt{3}, \sqrt{7})\), \(\Q(\sqrt{2}, \sqrt{-21})\), \(\Q(\sqrt{-2}, \sqrt{-21})\), \(\Q(\sqrt{3}, \sqrt{-7})\), \(\Q(\sqrt{-3}, \sqrt{7})\), \(\Q(\sqrt{2}, \sqrt{-7})\), \(\Q(\sqrt{-2}, \sqrt{7})\), \(\Q(\sqrt{-3}, \sqrt{-14})\), \(\Q(\sqrt{3}, \sqrt{-14})\), \(\Q(\sqrt{2}, \sqrt{7})\), \(\Q(\sqrt{-2}, \sqrt{-7})\), \(\Q(\sqrt{3}, \sqrt{14})\), \(\Q(\sqrt{-3}, \sqrt{14})\), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\sqrt{-2}, \sqrt{3})\), \(\Q(\sqrt{-6}, \sqrt{-7})\), \(\Q(\sqrt{-6}, \sqrt{7})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\sqrt{-2}, \sqrt{-3})\), \(\Q(\sqrt{6}, \sqrt{7})\), \(\Q(\sqrt{6}, \sqrt{-7})\), 4.4.903168.2, 4.0.903168.5, \(\Q(\zeta_{16})^+\), 4.0.2048.2, 4.0.18432.2, 4.4.18432.1, 4.0.100352.5, 4.4.100352.1, 8.0.12745506816.1, 8.0.12745506816.8, 8.0.49787136.1, 8.0.157351936.1, 8.0.12745506816.9, \(\Q(\zeta_{24})\), 8.0.12745506816.7, 8.0.796594176.2, 8.0.12745506816.3, 8.8.12745506816.1, 8.0.796594176.1, 8.0.12745506816.6, 8.0.12745506816.5, 8.0.12745506816.4, 8.0.12745506816.2, 8.0.3262849744896.7, \(\Q(\zeta_{16})\), 8.0.1358954496.4, 8.0.40282095616.2, 8.8.815712436224.1, 8.0.815712436224.4, 8.0.815712436224.1, 8.8.815712436224.2, 8.0.3262849744896.4, 8.0.3262849744896.5, 8.0.3262849744896.1, 8.0.3262849744896.2, 8.0.815712436224.2, 8.0.815712436224.3, 8.0.10070523904.2, 8.0.10070523904.1, 8.8.3262849744896.1, 8.0.3262849744896.6, 8.8.40282095616.1, 8.0.40282095616.1, 8.0.815712436224.5, 8.0.815712436224.6, 8.0.339738624.2, 8.0.339738624.1, 8.8.3262849744896.2, 8.0.3262849744896.3, \(\Q(\zeta_{48})^+\), 8.0.1358954496.3, 16.0.162447943996702457856.1, 16.0.10646188457767892278050816.4, 16.0.10646188457767892278050816.5, 16.0.10646188457767892278050816.8, 16.0.1622647227216566419456.1, 16.0.10646188457767892278050816.7, \(\Q(\zeta_{48})\), 16.0.665386778610493267378176.2, 16.0.665386778610493267378176.1, 16.16.10646188457767892278050816.1, 16.0.10646188457767892278050816.3, 16.0.10646188457767892278050816.9, 16.0.10646188457767892278050816.2, 16.0.10646188457767892278050816.6, 16.0.10646188457767892278050816.1

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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{8}$ 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

$p$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}$
$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}$