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Properties

Label 16.8.13315591412...3248.3
Degree $16$
Signature $[8, 4]$
Discriminant $2^{8}\cdot 17^{15}\cdot 6529^{4}$
Root discriminant $181.04$
Ramified primes $2, 17, 6529$
Class number $8$ (GRH)
Class group $[2, 2, 2]$ (GRH)
Galois group $(C_2\times OD_{16}).D_4$ (as 16T591)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![105165028996897, 2296056514362, -9010729094910, 1046870527544, -165662813949, -14293079195, 9801720863, -633327585, 85092020, 3543996, -2563925, 119471, -24059, 58, 111, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 + 111*x^14 + 58*x^13 - 24059*x^12 + 119471*x^11 - 2563925*x^10 + 3543996*x^9 + 85092020*x^8 - 633327585*x^7 + 9801720863*x^6 - 14293079195*x^5 - 165662813949*x^4 + 1046870527544*x^3 - 9010729094910*x^2 + 2296056514362*x + 105165028996897)
 
gp: K = bnfinit(x^16 - 3*x^15 + 111*x^14 + 58*x^13 - 24059*x^12 + 119471*x^11 - 2563925*x^10 + 3543996*x^9 + 85092020*x^8 - 633327585*x^7 + 9801720863*x^6 - 14293079195*x^5 - 165662813949*x^4 + 1046870527544*x^3 - 9010729094910*x^2 + 2296056514362*x + 105165028996897, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} + 111 x^{14} + 58 x^{13} - 24059 x^{12} + 119471 x^{11} - 2563925 x^{10} + 3543996 x^{9} + 85092020 x^{8} - 633327585 x^{7} + 9801720863 x^{6} - 14293079195 x^{5} - 165662813949 x^{4} + 1046870527544 x^{3} - 9010729094910 x^{2} + 2296056514362 x + 105165028996897 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1331559141233168225818951099338453248=2^{8}\cdot 17^{15}\cdot 6529^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $181.04$
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:  $2, 17, 6529$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{1218334639322009057862905852848624471862246448038514521831543879173514871053880669466263459606256999} a^{15} - \frac{414689033884529414984407887952859953638417490028841145873329673534220723552414503173246467638374304}{1218334639322009057862905852848624471862246448038514521831543879173514871053880669466263459606256999} a^{14} - \frac{321768865120284931569271292819688544916523005742127806578695909099737797481857138339430351652060722}{1218334639322009057862905852848624471862246448038514521831543879173514871053880669466263459606256999} a^{13} + \frac{403339985825270172176207634580700043471446417496343087885018512435777180505987205450799262716896736}{1218334639322009057862905852848624471862246448038514521831543879173514871053880669466263459606256999} a^{12} + \frac{99230036040590459979699001136720943153114732311902797441667821781672514988597957956454201619630406}{1218334639322009057862905852848624471862246448038514521831543879173514871053880669466263459606256999} a^{11} + \frac{13053902656329516506913685997352789255873638279859546824300127526931880848319195090667420979802678}{1218334639322009057862905852848624471862246448038514521831543879173514871053880669466263459606256999} a^{10} - \frac{210656422324470029772231503021189440212569251744576414389533493179893931883934501269835013300863078}{1218334639322009057862905852848624471862246448038514521831543879173514871053880669466263459606256999} a^{9} - \frac{472945963759612235429195297380683015453660156108923649172223311655964757154777868697099687772575963}{1218334639322009057862905852848624471862246448038514521831543879173514871053880669466263459606256999} a^{8} + \frac{449577929734180780489659717403555456526881004591167493433731779658010003439234684061591144638428723}{1218334639322009057862905852848624471862246448038514521831543879173514871053880669466263459606256999} a^{7} - \frac{229420577895661231738833084040954527112872418256831498281513817923486533521992043500655725739341358}{1218334639322009057862905852848624471862246448038514521831543879173514871053880669466263459606256999} a^{6} - \frac{131906813497493054130056964836586020919585277471438540336629555159469965762763785438876920205250414}{1218334639322009057862905852848624471862246448038514521831543879173514871053880669466263459606256999} a^{5} + \frac{215961551946247035180017481110599830648361172833557678344914402574670286758636703977299024643567332}{1218334639322009057862905852848624471862246448038514521831543879173514871053880669466263459606256999} a^{4} - \frac{260538589804730883250899804360263503510921831234562296743926371301872944249791289457226120729602475}{1218334639322009057862905852848624471862246448038514521831543879173514871053880669466263459606256999} a^{3} + \frac{156688242010759413500599007046456733339654150064975979074046702388804696931075731242351951322652119}{1218334639322009057862905852848624471862246448038514521831543879173514871053880669466263459606256999} a^{2} - \frac{400546287064245946378685570252694862211861612325692420085363465585911595813347487201103739668099381}{1218334639322009057862905852848624471862246448038514521831543879173514871053880669466263459606256999} a + \frac{386579886088529531879285073352169576603381135562305942172241734811189959410786992276643720448912432}{1218334639322009057862905852848624471862246448038514521831543879173514871053880669466263459606256999}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}$, which has order $8$ (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:  $11$
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:  \( 118491171291 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times OD_{16}).D_4$ (as 16T591):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 28 conjugacy class representatives for $(C_2\times OD_{16}).D_4$
Character table for $(C_2\times OD_{16}).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1, \(\Q(\zeta_{17})^+\)

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 16 siblings: data not computed
Degree 32 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$2$2.8.8.4$x^{8} + 2 x^{7} + 2 x^{6} + 8 x^{3} + 48$$2$$4$$8$$C_8$$[2]^{4}$
2.8.0.1$x^{8} + x^{4} + x^{3} + x + 1$$1$$8$$0$$C_8$$[\ ]^{8}$
17Data not computed
6529Data not computed