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Properties

Label 16.0.79377267773...0761.3
Degree $16$
Signature $[0, 8]$
Discriminant $13^{14}\cdot 17^{10}$
Root discriminant $55.43$
Ramified primes $13, 17$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $(C_2\times C_4).D_4$ (as 16T121)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![32353, -101281, 141895, -97761, 95090, -69904, 49941, -28468, 15431, -7717, 4080, -1862, 735, -224, 52, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 52*x^14 - 224*x^13 + 735*x^12 - 1862*x^11 + 4080*x^10 - 7717*x^9 + 15431*x^8 - 28468*x^7 + 49941*x^6 - 69904*x^5 + 95090*x^4 - 97761*x^3 + 141895*x^2 - 101281*x + 32353)
 
gp: K = bnfinit(x^16 - 8*x^15 + 52*x^14 - 224*x^13 + 735*x^12 - 1862*x^11 + 4080*x^10 - 7717*x^9 + 15431*x^8 - 28468*x^7 + 49941*x^6 - 69904*x^5 + 95090*x^4 - 97761*x^3 + 141895*x^2 - 101281*x + 32353, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 52 x^{14} - 224 x^{13} + 735 x^{12} - 1862 x^{11} + 4080 x^{10} - 7717 x^{9} + 15431 x^{8} - 28468 x^{7} + 49941 x^{6} - 69904 x^{5} + 95090 x^{4} - 97761 x^{3} + 141895 x^{2} - 101281 x + 32353 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(7937726777341695855516080761=13^{14}\cdot 17^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $55.43$
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:  $13, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{99} a^{12} - \frac{2}{33} a^{11} + \frac{29}{99} a^{10} + \frac{1}{11} a^{9} + \frac{43}{99} a^{8} + \frac{5}{99} a^{7} + \frac{16}{33} a^{6} - \frac{28}{99} a^{5} - \frac{29}{99} a^{4} + \frac{10}{99} a^{3} - \frac{43}{99} a^{2} - \frac{13}{33} a + \frac{46}{99}$, $\frac{1}{99} a^{13} - \frac{7}{99} a^{11} - \frac{5}{33} a^{10} - \frac{2}{99} a^{9} - \frac{34}{99} a^{8} - \frac{7}{33} a^{7} - \frac{37}{99} a^{6} + \frac{1}{99} a^{5} + \frac{34}{99} a^{4} + \frac{17}{99} a^{3} + \frac{10}{99} a - \frac{7}{33}$, $\frac{1}{94464744422301} a^{14} - \frac{7}{94464744422301} a^{13} + \frac{408489527551}{94464744422301} a^{12} - \frac{222812469565}{8587704038391} a^{11} - \frac{34722364832137}{94464744422301} a^{10} + \frac{7149259330387}{94464744422301} a^{9} - \frac{8682202026529}{31488248140767} a^{8} + \frac{11443519839649}{31488248140767} a^{7} - \frac{11859076270543}{94464744422301} a^{6} + \frac{10193366268925}{94464744422301} a^{5} + \frac{443391869711}{1166231412621} a^{4} - \frac{137053373977}{954189337599} a^{3} + \frac{35896228567871}{94464744422301} a^{2} - \frac{35245376289061}{94464744422301} a + \frac{3546165347249}{94464744422301}$, $\frac{1}{778672888273027143} a^{15} + \frac{374}{70788444388457013} a^{14} + \frac{1933596087474278}{778672888273027143} a^{13} - \frac{111829713745934}{28839736602704709} a^{12} - \frac{60583024701542282}{778672888273027143} a^{11} + \frac{241330542224602940}{778672888273027143} a^{10} - \frac{373243299644383948}{778672888273027143} a^{9} + \frac{67659215775769514}{259557629424342381} a^{8} + \frac{18443446440714262}{70788444388457013} a^{7} + \frac{65357312144636720}{778672888273027143} a^{6} - \frac{24931772470387294}{778672888273027143} a^{5} + \frac{19894226197505560}{86519209808114127} a^{4} + \frac{267850722712693316}{778672888273027143} a^{3} - \frac{124404831211228048}{259557629424342381} a^{2} + \frac{2833702499588581}{259557629424342381} a - \frac{180355759955590541}{778672888273027143}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  $7$
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:  \( 3583174.458 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times C_4).D_4$ (as 16T121):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 28 conjugacy class representatives for $(C_2\times C_4).D_4$
Character table for $(C_2\times C_4).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{13}) \), 4.4.37349.1, 4.0.634933.1, 4.0.2873.1, 8.4.89093921102069.1, 8.4.308283464021.2, 8.0.403139914489.3

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
13Data not computed
$17$17.4.3.3$x^{4} + 51$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.3$x^{4} + 51$$4$$1$$3$$C_4$$[\ ]_{4}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$