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Properties

Label 32.0.79407184549...0000.1
Degree $32$
Signature $[0, 16]$
Discriminant $2^{88}\cdot 3^{16}\cdot 5^{24}$
Root discriminant $38.96$
Ramified primes $2, 3, 5$
Class number $100$ (GRH)
Class group $[5, 20]$ (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![256, 0, -512, 0, 128, 0, 1280, 0, -2560, 0, 4352, 0, -1056, 0, -10880, 0, 21744, 0, -5440, 0, -264, 0, 544, 0, -160, 0, 40, 0, 2, 0, -4, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 - 4*x^30 + 2*x^28 + 40*x^26 - 160*x^24 + 544*x^22 - 264*x^20 - 5440*x^18 + 21744*x^16 - 10880*x^14 - 1056*x^12 + 4352*x^10 - 2560*x^8 + 1280*x^6 + 128*x^4 - 512*x^2 + 256)
 
gp: K = bnfinit(x^32 - 4*x^30 + 2*x^28 + 40*x^26 - 160*x^24 + 544*x^22 - 264*x^20 - 5440*x^18 + 21744*x^16 - 10880*x^14 - 1056*x^12 + 4352*x^10 - 2560*x^8 + 1280*x^6 + 128*x^4 - 512*x^2 + 256, 1)
 

Normalized defining polynomial

\( x^{32} - 4 x^{30} + 2 x^{28} + 40 x^{26} - 160 x^{24} + 544 x^{22} - 264 x^{20} - 5440 x^{18} + 21744 x^{16} - 10880 x^{14} - 1056 x^{12} + 4352 x^{10} - 2560 x^{8} + 1280 x^{6} + 128 x^{4} - 512 x^{2} + 256 \)

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:  \(794071845499378503449051136000000000000000000000000=2^{88}\cdot 3^{16}\cdot 5^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.96$
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$$\chi_{240}(1,·)$, $\chi_{240}(133,·)$, $\chi_{240}(137,·)$, $\chi_{240}(13,·)$, $\chi_{240}(17,·)$, $\chi_{240}(149,·)$, $\chi_{240}(89,·)$, $\chi_{240}(29,·)$, $\chi_{240}(161,·)$, $\chi_{240}(37,·)$, $\chi_{240}(41,·)$, $\chi_{240}(173,·)$, $\chi_{240}(157,·)$, $\chi_{240}(49,·)$, $\chi_{240}(181,·)$, $\chi_{240}(61,·)$, $\chi_{240}(53,·)$, $\chi_{240}(193,·)$, $\chi_{240}(197,·)$, $\chi_{240}(73,·)$, $\chi_{240}(77,·)$, $\chi_{240}(209,·)$, $\chi_{240}(217,·)$, $\chi_{240}(221,·)$, $\chi_{240}(101,·)$, $\chi_{240}(97,·)$, $\chi_{240}(229,·)$, $\chi_{240}(233,·)$, $\chi_{240}(109,·)$, $\chi_{240}(113,·)$, $\chi_{240}(169,·)$, $\chi_{240}(121,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{4} a^{8}$, $\frac{1}{4} a^{9}$, $\frac{1}{4} a^{10}$, $\frac{1}{4} a^{11}$, $\frac{1}{8} a^{12}$, $\frac{1}{8} a^{13}$, $\frac{1}{8} a^{14}$, $\frac{1}{8} a^{15}$, $\frac{1}{16} a^{16}$, $\frac{1}{16} a^{17}$, $\frac{1}{16} a^{18}$, $\frac{1}{16} a^{19}$, $\frac{1}{1312} a^{20} - \frac{3}{41} a^{10} + \frac{1}{41}$, $\frac{1}{1312} a^{21} - \frac{3}{41} a^{11} + \frac{1}{41} a$, $\frac{1}{1312} a^{22} + \frac{17}{328} a^{12} + \frac{1}{41} a^{2}$, $\frac{1}{1312} a^{23} + \frac{17}{328} a^{13} + \frac{1}{41} a^{3}$, $\frac{1}{18368} a^{24} + \frac{1}{56} a^{18} + \frac{5}{164} a^{14} - \frac{1}{56} a^{12} + \frac{1}{7} a^{6} + \frac{3}{41} a^{4} - \frac{1}{7}$, $\frac{1}{18368} a^{25} + \frac{1}{56} a^{19} + \frac{5}{164} a^{15} - \frac{1}{56} a^{13} + \frac{1}{7} a^{7} + \frac{3}{41} a^{5} - \frac{1}{7} a$, $\frac{1}{17651648} a^{26} - \frac{25}{4412912} a^{24} + \frac{165}{1260832} a^{22} + \frac{2803}{8825824} a^{20} - \frac{165}{107632} a^{18} + \frac{825}{157604} a^{16} - \frac{2803}{1103228} a^{14} - \frac{165}{551614} a^{12} - \frac{11371}{157604} a^{10} - \frac{2803}{26908} a^{8} + \frac{165}{551614} a^{6} + \frac{2803}{78802} a^{4} - \frac{34}{275807} a^{2} - \frac{16221}{275807}$, $\frac{1}{17651648} a^{27} - \frac{25}{4412912} a^{25} + \frac{165}{1260832} a^{23} + \frac{2803}{8825824} a^{21} - \frac{165}{107632} a^{19} + \frac{825}{157604} a^{17} - \frac{2803}{1103228} a^{15} - \frac{165}{551614} a^{13} - \frac{11371}{157604} a^{11} - \frac{2803}{26908} a^{9} + \frac{165}{551614} a^{7} + \frac{2803}{78802} a^{5} - \frac{34}{275807} a^{3} - \frac{16221}{275807} a$, $\frac{1}{35303296} a^{28} - \frac{1}{17651648} a^{24} + \frac{493}{2206456} a^{22} - \frac{165}{1260832} a^{20} - \frac{2873}{1103228} a^{18} + \frac{563}{53816} a^{16} - \frac{825}{157604} a^{14} - \frac{1121}{2206456} a^{12} - \frac{19618}{275807} a^{10} + \frac{3283}{78802} a^{8} - \frac{981}{6727} a^{6} + \frac{19618}{275807} a^{4} - \frac{9570}{39401} a^{2} + \frac{34}{275807}$, $\frac{1}{35303296} a^{29} - \frac{1}{17651648} a^{25} + \frac{493}{2206456} a^{23} - \frac{165}{1260832} a^{21} - \frac{2873}{1103228} a^{19} + \frac{563}{53816} a^{17} - \frac{825}{157604} a^{15} - \frac{1121}{2206456} a^{13} - \frac{19618}{275807} a^{11} + \frac{3283}{78802} a^{9} - \frac{981}{6727} a^{7} + \frac{19618}{275807} a^{5} - \frac{9570}{39401} a^{3} + \frac{34}{275807} a$, $\frac{1}{35303296} a^{30} - \frac{114243}{275807}$, $\frac{1}{35303296} a^{31} - \frac{114243}{275807} a$

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

Class group and class number

$C_{5}\times C_{20}$, which has order $100$ (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:  \( -\frac{1479}{551614} a^{30} + \frac{1479}{157604} a^{28} + \frac{51}{2206456} a^{26} - \frac{1479}{13454} a^{24} + \frac{14790}{39401} a^{22} - \frac{343128}{275807} a^{20} - \frac{5916}{275807} a^{18} + \frac{9424999}{630416} a^{16} - \frac{343128}{6727} a^{14} + \frac{5916}{275807} a^{12} + \frac{686256}{39401} a^{10} - \frac{2827848}{275807} a^{8} + \frac{1656276}{275807} a^{6} - \frac{567936}{275807} a^{2} + \frac{47328}{39401} \) (order $30$)
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:  \( 549552630502.74054 \) (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

\(\Q(\sqrt{-6}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-30}) \), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\zeta_{16})^+\), 4.0.18432.2, \(\Q(\sqrt{-6}, \sqrt{10})\), \(\Q(\sqrt{5}, \sqrt{-6})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{2}, \sqrt{-15})\), 4.4.51200.1, 4.0.460800.2, \(\Q(\sqrt{-3}, \sqrt{10})\), \(\Q(\sqrt{-3}, \sqrt{5})\), 4.4.2304000.2, 4.0.256000.4, 4.4.2304000.1, 4.0.256000.2, \(\Q(\zeta_{15})^+\), 4.0.8000.2, 4.4.72000.1, \(\Q(\zeta_{5})\), 8.0.339738624.2, 8.0.207360000.1, 8.0.212336640000.7, 8.8.2621440000.1, 8.0.212336640000.4, 8.0.212336640000.5, 8.0.212336640000.2, 8.0.5308416000000.8, 8.0.5308416000000.4, 8.0.5184000000.1, 8.0.5184000000.5, 8.8.5308416000000.1, 8.0.65536000000.1, 8.8.5184000000.1, 8.0.64000000.2, 8.0.5308416000000.5, 8.0.5308416000000.1, \(\Q(\zeta_{15})\), 8.0.5184000000.3, 16.0.45086848686489600000000.1, 16.0.28179280429056000000000000.2, 16.0.26873856000000000000.2, 16.16.28179280429056000000000000.2, 16.0.4294967296000000000000.1, 16.0.28179280429056000000000000.5, 16.0.28179280429056000000000000.4

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{32}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{8}$

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