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Properties

Label 32.0.87742381336...1296.1
Degree $32$
Signature $[0, 16]$
Discriminant $2^{88}\cdot 17^{28}$
Root discriminant $80.26$
Ramified primes $2, 17$
Class number $579904$ (GRH)
Class group $[8, 72488]$ (GRH)
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![256, 0, 18432, 0, 327168, 0, 2526720, 0, 10286208, 0, 24884992, 0, 38567232, 0, 40097856, 0, 28730096, 0, 14374848, 0, 5034128, 0, 1224784, 0, 203404, 0, 22360, 0, 1542, 0, 60, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^32 + 60*x^30 + 1542*x^28 + 22360*x^26 + 203404*x^24 + 1224784*x^22 + 5034128*x^20 + 14374848*x^18 + 28730096*x^16 + 40097856*x^14 + 38567232*x^12 + 24884992*x^10 + 10286208*x^8 + 2526720*x^6 + 327168*x^4 + 18432*x^2 + 256)
 
gp: K = bnfinit(x^32 + 60*x^30 + 1542*x^28 + 22360*x^26 + 203404*x^24 + 1224784*x^22 + 5034128*x^20 + 14374848*x^18 + 28730096*x^16 + 40097856*x^14 + 38567232*x^12 + 24884992*x^10 + 10286208*x^8 + 2526720*x^6 + 327168*x^4 + 18432*x^2 + 256, 1)
 

Normalized defining polynomial

\( x^{32} + 60 x^{30} + 1542 x^{28} + 22360 x^{26} + 203404 x^{24} + 1224784 x^{22} + 5034128 x^{20} + 14374848 x^{18} + 28730096 x^{16} + 40097856 x^{14} + 38567232 x^{12} + 24884992 x^{10} + 10286208 x^{8} + 2526720 x^{6} + 327168 x^{4} + 18432 x^{2} + 256 \)

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:  \(8774238133649262699883985224531159760379882962200337418551296=2^{88}\cdot 17^{28}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $80.26$
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:  $2, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(272=2^{4}\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{272}(1,·)$, $\chi_{272}(259,·)$, $\chi_{272}(257,·)$, $\chi_{272}(9,·)$, $\chi_{272}(145,·)$, $\chi_{272}(19,·)$, $\chi_{272}(89,·)$, $\chi_{272}(25,·)$, $\chi_{272}(155,·)$, $\chi_{272}(161,·)$, $\chi_{272}(35,·)$, $\chi_{272}(169,·)$, $\chi_{272}(171,·)$, $\chi_{272}(49,·)$, $\chi_{272}(179,·)$, $\chi_{272}(137,·)$, $\chi_{272}(185,·)$, $\chi_{272}(59,·)$, $\chi_{272}(67,·)$, $\chi_{272}(33,·)$, $\chi_{272}(203,·)$, $\chi_{272}(81,·)$, $\chi_{272}(83,·)$, $\chi_{272}(43,·)$, $\chi_{272}(217,·)$, $\chi_{272}(219,·)$, $\chi_{272}(225,·)$, $\chi_{272}(123,·)$, $\chi_{272}(195,·)$, $\chi_{272}(115,·)$, $\chi_{272}(121,·)$, $\chi_{272}(251,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{4} a^{8}$, $\frac{1}{4} a^{9}$, $\frac{1}{4} a^{10}$, $\frac{1}{4} a^{11}$, $\frac{1}{8} a^{12}$, $\frac{1}{8} a^{13}$, $\frac{1}{8} a^{14}$, $\frac{1}{8} a^{15}$, $\frac{1}{16} a^{16}$, $\frac{1}{16} a^{17}$, $\frac{1}{16} a^{18}$, $\frac{1}{16} a^{19}$, $\frac{1}{32} a^{20}$, $\frac{1}{32} a^{21}$, $\frac{1}{32} a^{22}$, $\frac{1}{32} a^{23}$, $\frac{1}{64} a^{24}$, $\frac{1}{64} a^{25}$, $\frac{1}{64} a^{26}$, $\frac{1}{64} a^{27}$, $\frac{1}{128} a^{28}$, $\frac{1}{128} a^{29}$, $\frac{1}{81786316286314649278336} a^{30} + \frac{115257091883288870435}{81786316286314649278336} a^{28} - \frac{162925060517792049881}{40893158143157324639168} a^{26} - \frac{143480371513740286843}{40893158143157324639168} a^{24} - \frac{93897984886449722031}{10223289535789331159792} a^{22} - \frac{194679272284885524673}{20446579071578662319584} a^{20} - \frac{157791301553184174681}{5111644767894665579896} a^{18} + \frac{171658091979459332379}{10223289535789331159792} a^{16} + \frac{132090643030447190705}{2555822383947332789948} a^{14} + \frac{203995289424305691375}{5111644767894665579896} a^{12} + \frac{141516308810630617407}{1277911191973666394974} a^{10} - \frac{156740116938400618071}{1277911191973666394974} a^{8} - \frac{45328647347476228489}{638955595986833197487} a^{6} + \frac{161564535550950583881}{1277911191973666394974} a^{4} + \frac{68017781521262364200}{638955595986833197487} a^{2} - \frac{94108401705839125592}{638955595986833197487}$, $\frac{1}{81786316286314649278336} a^{31} + \frac{115257091883288870435}{81786316286314649278336} a^{29} - \frac{162925060517792049881}{40893158143157324639168} a^{27} - \frac{143480371513740286843}{40893158143157324639168} a^{25} - \frac{93897984886449722031}{10223289535789331159792} a^{23} - \frac{194679272284885524673}{20446579071578662319584} a^{21} - \frac{157791301553184174681}{5111644767894665579896} a^{19} + \frac{171658091979459332379}{10223289535789331159792} a^{17} + \frac{132090643030447190705}{2555822383947332789948} a^{15} + \frac{203995289424305691375}{5111644767894665579896} a^{13} + \frac{141516308810630617407}{1277911191973666394974} a^{11} - \frac{156740116938400618071}{1277911191973666394974} a^{9} - \frac{45328647347476228489}{638955595986833197487} a^{7} + \frac{161564535550950583881}{1277911191973666394974} a^{5} + \frac{68017781521262364200}{638955595986833197487} a^{3} - \frac{94108401705839125592}{638955595986833197487} a$

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

Class group and class number

$C_{8}\times C_{72488}$, which has order $579904$ (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:  \( -1 \) (order $2$)
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:  \( 303826092469.64355 \) (assuming GRH)
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

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{34}) \), 4.0.2048.2, \(\Q(\sqrt{2}, \sqrt{17})\), 4.0.591872.5, 4.4.314432.1, 4.4.4913.1, 4.0.10061824.1, 4.0.10061824.2, 8.0.350312464384.1, 8.8.98867482624.1, 8.0.101240302206976.1, 8.0.1721085137518592.2, 8.0.1721085137518592.1, \(\Q(\zeta_{17})^+\), 8.8.1680747204608.1, 16.0.10249598790959829536343064576.2, 16.0.2962134050587390736003145662464.1, 16.16.2824911165797606216433664.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{16}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{8}$

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

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
$17$17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$