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Properties

Label 32.0.42127744383...0000.3
Degree $32$
Signature $[0, 16]$
Discriminant $2^{124}\cdot 5^{24}\cdot 7^{16}$
Root discriminant $129.80$
Ramified primes $2, 5, 7$
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![16, 0, 2944, 0, 108576, 0, 1684032, 0, 13396712, 0, 58587840, 0, 144210848, 0, 205236512, 0, 175529272, 0, 93050912, 0, 31193320, 0, 6660000, 0, 899720, 0, 75184, 0, 3696, 0, 96, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 96*x^30 + 3696*x^28 + 75184*x^26 + 899720*x^24 + 6660000*x^22 + 31193320*x^20 + 93050912*x^18 + 175529272*x^16 + 205236512*x^14 + 144210848*x^12 + 58587840*x^10 + 13396712*x^8 + 1684032*x^6 + 108576*x^4 + 2944*x^2 + 16)
 
gp: K = bnfinit(x^32 + 96*x^30 + 3696*x^28 + 75184*x^26 + 899720*x^24 + 6660000*x^22 + 31193320*x^20 + 93050912*x^18 + 175529272*x^16 + 205236512*x^14 + 144210848*x^12 + 58587840*x^10 + 13396712*x^8 + 1684032*x^6 + 108576*x^4 + 2944*x^2 + 16, 1)
 

Normalized defining polynomial

\( x^{32} + 96 x^{30} + 3696 x^{28} + 75184 x^{26} + 899720 x^{24} + 6660000 x^{22} + 31193320 x^{20} + 93050912 x^{18} + 175529272 x^{16} + 205236512 x^{14} + 144210848 x^{12} + 58587840 x^{10} + 13396712 x^{8} + 1684032 x^{6} + 108576 x^{4} + 2944 x^{2} + 16 \)

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:  \(42127744383897781264481528176959874165374976000000000000000000000000=2^{124}\cdot 5^{24}\cdot 7^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $129.80$
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:  $2, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1120=2^{5}\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{1120}(1,·)$, $\chi_{1120}(909,·)$, $\chi_{1120}(657,·)$, $\chi_{1120}(533,·)$, $\chi_{1120}(281,·)$, $\chi_{1120}(153,·)$, $\chi_{1120}(1093,·)$, $\chi_{1120}(197,·)$, $\chi_{1120}(561,·)$, $\chi_{1120}(169,·)$, $\chi_{1120}(813,·)$, $\chi_{1120}(349,·)$, $\chi_{1120}(433,·)$, $\chi_{1120}(181,·)$, $\chi_{1120}(841,·)$, $\chi_{1120}(629,·)$, $\chi_{1120}(449,·)$, $\chi_{1120}(69,·)$, $\chi_{1120}(97,·)$, $\chi_{1120}(713,·)$, $\chi_{1120}(461,·)$, $\chi_{1120}(1037,·)$, $\chi_{1120}(729,·)$, $\chi_{1120}(477,·)$, $\chi_{1120}(993,·)$, $\chi_{1120}(741,·)$, $\chi_{1120}(253,·)$, $\chi_{1120}(1009,·)$, $\chi_{1120}(757,·)$, $\chi_{1120}(937,·)$, $\chi_{1120}(377,·)$, $\chi_{1120}(1021,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8}$, $\frac{1}{2} a^{9}$, $\frac{1}{2} a^{10}$, $\frac{1}{2} a^{11}$, $\frac{1}{2} a^{12}$, $\frac{1}{2} a^{13}$, $\frac{1}{2} a^{14}$, $\frac{1}{2} a^{15}$, $\frac{1}{4} a^{16}$, $\frac{1}{4} a^{17}$, $\frac{1}{4} a^{18}$, $\frac{1}{4} a^{19}$, $\frac{1}{4} a^{20}$, $\frac{1}{4} a^{21}$, $\frac{1}{4} a^{22}$, $\frac{1}{4} a^{23}$, $\frac{1}{152} a^{24} - \frac{7}{76} a^{22} + \frac{3}{76} a^{20} + \frac{1}{76} a^{18} + \frac{1}{19} a^{16} - \frac{5}{38} a^{14} - \frac{1}{38} a^{12} + \frac{1}{19} a^{10} - \frac{4}{19} a^{8} - \frac{7}{19} a^{6} - \frac{8}{19} a^{4} + \frac{8}{19} a^{2} - \frac{3}{19}$, $\frac{1}{152} a^{25} - \frac{7}{76} a^{23} + \frac{3}{76} a^{21} + \frac{1}{76} a^{19} + \frac{1}{19} a^{17} - \frac{5}{38} a^{15} - \frac{1}{38} a^{13} + \frac{1}{19} a^{11} - \frac{4}{19} a^{9} - \frac{7}{19} a^{7} - \frac{8}{19} a^{5} + \frac{8}{19} a^{3} - \frac{3}{19} a$, $\frac{1}{152} a^{26} + \frac{5}{76} a^{20} - \frac{1}{76} a^{18} + \frac{2}{19} a^{16} + \frac{5}{38} a^{14} + \frac{7}{38} a^{12} + \frac{1}{38} a^{10} + \frac{7}{38} a^{8} + \frac{8}{19} a^{6} - \frac{9}{19} a^{4} - \frac{5}{19} a^{2} - \frac{4}{19}$, $\frac{1}{152} a^{27} + \frac{5}{76} a^{21} - \frac{1}{76} a^{19} + \frac{2}{19} a^{17} + \frac{5}{38} a^{15} + \frac{7}{38} a^{13} + \frac{1}{38} a^{11} + \frac{7}{38} a^{9} + \frac{8}{19} a^{7} - \frac{9}{19} a^{5} - \frac{5}{19} a^{3} - \frac{4}{19} a$, $\frac{1}{1064} a^{28} - \frac{3}{1064} a^{26} + \frac{1}{1064} a^{24} + \frac{9}{133} a^{22} - \frac{8}{133} a^{20} - \frac{45}{532} a^{18} - \frac{29}{532} a^{16} + \frac{3}{133} a^{14} + \frac{55}{266} a^{12} + \frac{3}{133} a^{10} + \frac{3}{133} a^{8} + \frac{36}{133} a^{6} + \frac{2}{19} a^{4} + \frac{1}{7} a^{2} - \frac{48}{133}$, $\frac{1}{1064} a^{29} - \frac{3}{1064} a^{27} + \frac{1}{1064} a^{25} + \frac{9}{133} a^{23} - \frac{8}{133} a^{21} - \frac{45}{532} a^{19} - \frac{29}{532} a^{17} + \frac{3}{133} a^{15} + \frac{55}{266} a^{13} + \frac{3}{133} a^{11} + \frac{3}{133} a^{9} + \frac{36}{133} a^{7} + \frac{2}{19} a^{5} + \frac{1}{7} a^{3} - \frac{48}{133} a$, $\frac{1}{796200721358242640706900334940680364504} a^{30} - \frac{262234743796482075417812578072840865}{796200721358242640706900334940680364504} a^{28} + \frac{2300443659195794722563731163487105489}{796200721358242640706900334940680364504} a^{26} + \frac{377186939415773454002740807199491151}{398100360679121320353450167470340182252} a^{24} - \frac{11861322613122273633881612977562726579}{199050180339560660176725083735170091126} a^{22} - \frac{33349248910197627991135178527095197335}{398100360679121320353450167470340182252} a^{20} + \frac{1734758833926166602191072024209462373}{14217870024254332869766077409655006509} a^{18} - \frac{21110360239552040196139728673899329577}{398100360679121320353450167470340182252} a^{16} + \frac{14227650723586514509941324843518204253}{199050180339560660176725083735170091126} a^{14} + \frac{27826911702119284769707948908242736}{5238162640514754215176975887767633977} a^{12} + \frac{1031817689368216615917281697832110760}{5238162640514754215176975887767633977} a^{10} - \frac{2081696693760338371068761979978418113}{28435740048508665739532154819310013018} a^{8} - \frac{41227459770888533880652689028252642201}{99525090169780330088362541867585045563} a^{6} - \frac{20313724406195299986176282261049087849}{99525090169780330088362541867585045563} a^{4} - \frac{24605341680269188269021627328339823842}{99525090169780330088362541867585045563} a^{2} - \frac{42891769580875068954655880460909518106}{99525090169780330088362541867585045563}$, $\frac{1}{796200721358242640706900334940680364504} a^{31} - \frac{262234743796482075417812578072840865}{796200721358242640706900334940680364504} a^{29} + \frac{2300443659195794722563731163487105489}{796200721358242640706900334940680364504} a^{27} + \frac{377186939415773454002740807199491151}{398100360679121320353450167470340182252} a^{25} - \frac{11861322613122273633881612977562726579}{199050180339560660176725083735170091126} a^{23} - \frac{33349248910197627991135178527095197335}{398100360679121320353450167470340182252} a^{21} + \frac{1734758833926166602191072024209462373}{14217870024254332869766077409655006509} a^{19} - \frac{21110360239552040196139728673899329577}{398100360679121320353450167470340182252} a^{17} + \frac{14227650723586514509941324843518204253}{199050180339560660176725083735170091126} a^{15} + \frac{27826911702119284769707948908242736}{5238162640514754215176975887767633977} a^{13} + \frac{1031817689368216615917281697832110760}{5238162640514754215176975887767633977} a^{11} - \frac{2081696693760338371068761979978418113}{28435740048508665739532154819310013018} a^{9} - \frac{41227459770888533880652689028252642201}{99525090169780330088362541867585045563} a^{7} - \frac{20313724406195299986176282261049087849}{99525090169780330088362541867585045563} a^{5} - \frac{24605341680269188269021627328339823842}{99525090169780330088362541867585045563} a^{3} - \frac{42891769580875068954655880460909518106}{99525090169780330088362541867585045563} a$

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

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{10}) \), 4.4.392000.1, \(\Q(\sqrt{2}, \sqrt{5})\), 4.4.6125.1, \(\Q(\zeta_{16})^+\), 4.4.51200.1, 4.4.12544000.2, 4.4.12544000.1, 8.8.153664000000.1, 8.8.2621440000.1, 8.8.157351936000000.4, 8.0.5156108238848.1, 8.0.3222567649280000.67, 8.0.33554432000000.1, 8.0.33554432000000.2, 16.16.24759631762948096000000000000.2, 16.0.10384942254186025084518400000000.5, 16.0.1125899906842624000000000000.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{4}$ R R ${\href{/LocalNumberField/11.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{4}$

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
5Data not computed
$7$7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$