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Properties

Label 16.0.73509189062...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{8}\cdot 5^{14}\cdot 19^{6}$
Root discriminant $17.44$
Ramified primes $2, 5, 19$
Class number $2$
Class group $[2]$
Galois group $(C_2\times Q_8).C_2^3$ (as 16T226)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 14, 78, 183, 126, -311, -415, 109, 524, -57, -240, 2, 66, -4, -3, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - 3*x^14 - 4*x^13 + 66*x^12 + 2*x^11 - 240*x^10 - 57*x^9 + 524*x^8 + 109*x^7 - 415*x^6 - 311*x^5 + 126*x^4 + 183*x^3 + 78*x^2 + 14*x + 1)
 
gp: K = bnfinit(x^16 - 3*x^15 - 3*x^14 - 4*x^13 + 66*x^12 + 2*x^11 - 240*x^10 - 57*x^9 + 524*x^8 + 109*x^7 - 415*x^6 - 311*x^5 + 126*x^4 + 183*x^3 + 78*x^2 + 14*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} - 3 x^{14} - 4 x^{13} + 66 x^{12} + 2 x^{11} - 240 x^{10} - 57 x^{9} + 524 x^{8} + 109 x^{7} - 415 x^{6} - 311 x^{5} + 126 x^{4} + 183 x^{3} + 78 x^{2} + 14 x + 1 \)

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:  \(73509189062500000000=2^{8}\cdot 5^{14}\cdot 19^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.44$
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:  $2, 5, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{49095746942951} a^{15} - \frac{12847270199914}{49095746942951} a^{14} + \frac{7261354535360}{49095746942951} a^{13} - \frac{17749533554213}{49095746942951} a^{12} - \frac{7831059844489}{49095746942951} a^{11} + \frac{5589105498230}{49095746942951} a^{10} - \frac{8972825909517}{49095746942951} a^{9} - \frac{21576097370601}{49095746942951} a^{8} + \frac{12665736782151}{49095746942951} a^{7} - \frac{13028706489397}{49095746942951} a^{6} + \frac{8318202300996}{49095746942951} a^{5} + \frac{6054276987850}{49095746942951} a^{4} - \frac{9024964333944}{49095746942951} a^{3} - \frac{6939948573659}{49095746942951} a^{2} - \frac{18670158081185}{49095746942951} a + \frac{14956066088080}{49095746942951}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( -\frac{28015239468195}{49095746942951} a^{15} + \frac{93213567852754}{49095746942951} a^{14} + \frac{52364982911857}{49095746942951} a^{13} + \frac{96815676902528}{49095746942951} a^{12} - \frac{1874164307085856}{49095746942951} a^{11} + \frac{569133062092902}{49095746942951} a^{10} + \frac{6477887056383491}{49095746942951} a^{9} - \frac{608487649619425}{49095746942951} a^{8} - \frac{14288904712281286}{49095746942951} a^{7} + \frac{1904191061375281}{49095746942951} a^{6} + \frac{10708996976472996}{49095746942951} a^{5} + \frac{4810979884815240}{49095746942951} a^{4} - \frac{5013344172947876}{49095746942951} a^{3} - \frac{3264818594512807}{49095746942951} a^{2} - \frac{1000074647161265}{49095746942951} a - \frac{61815470911378}{49095746942951} \) (order $10$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 5112.72299701 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times Q_8).C_2^3$ (as 16T226):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 23 conjugacy class representatives for $(C_2\times Q_8).C_2^3$
Character table for $(C_2\times Q_8).C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.2.475.1, \(\Q(\zeta_{5})\), 4.2.2375.1, 8.0.5640625.1

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 ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$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}$
$5$5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$19$19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$