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Properties

Label 16.0.10574772515...5625.6
Degree $16$
Signature $[0, 8]$
Discriminant $5^{10}\cdot 101^{8}$
Root discriminant $27.48$
Ramified primes $5, 101$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $C_2^4.D_4$ (as 16T339)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![355, -1290, 1310, 460, -1311, 689, 1220, -2222, 2055, -1513, 978, -592, 311, -126, 38, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 38*x^14 - 126*x^13 + 311*x^12 - 592*x^11 + 978*x^10 - 1513*x^9 + 2055*x^8 - 2222*x^7 + 1220*x^6 + 689*x^5 - 1311*x^4 + 460*x^3 + 1310*x^2 - 1290*x + 355)
 
gp: K = bnfinit(x^16 - 8*x^15 + 38*x^14 - 126*x^13 + 311*x^12 - 592*x^11 + 978*x^10 - 1513*x^9 + 2055*x^8 - 2222*x^7 + 1220*x^6 + 689*x^5 - 1311*x^4 + 460*x^3 + 1310*x^2 - 1290*x + 355, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 38 x^{14} - 126 x^{13} + 311 x^{12} - 592 x^{11} + 978 x^{10} - 1513 x^{9} + 2055 x^{8} - 2222 x^{7} + 1220 x^{6} + 689 x^{5} - 1311 x^{4} + 460 x^{3} + 1310 x^{2} - 1290 x + 355 \)

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:  \(105747725158992197265625=5^{10}\cdot 101^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.48$
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:  $5, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{11} a^{13} - \frac{1}{11} a^{12} + \frac{5}{11} a^{11} - \frac{5}{11} a^{9} - \frac{5}{11} a^{8} - \frac{5}{11} a^{7} - \frac{5}{11} a^{6} - \frac{1}{11} a^{5} - \frac{4}{11} a^{4} + \frac{2}{11} a^{3} + \frac{5}{11} a^{2} - \frac{4}{11} a + \frac{3}{11}$, $\frac{1}{170479561109} a^{14} - \frac{7}{170479561109} a^{13} + \frac{3905741913}{15498141919} a^{12} + \frac{83180156051}{170479561109} a^{11} + \frac{4542304509}{170479561109} a^{10} - \frac{32967723}{120993301} a^{9} + \frac{40615901736}{170479561109} a^{8} + \frac{5926099}{120993301} a^{7} - \frac{62901474134}{170479561109} a^{6} + \frac{38183211585}{170479561109} a^{5} - \frac{35689480075}{170479561109} a^{4} + \frac{56437011767}{170479561109} a^{3} - \frac{1909489363}{170479561109} a^{2} + \frac{43159906212}{170479561109} a + \frac{271361115}{2401120579}$, $\frac{1}{25742413727459} a^{15} + \frac{4}{1514259631027} a^{14} - \frac{623456941999}{25742413727459} a^{13} - \frac{5575018085311}{25742413727459} a^{12} - \frac{785844054588}{1980185671343} a^{11} - \frac{4990645077911}{25742413727459} a^{10} + \frac{4832759558457}{25742413727459} a^{9} + \frac{1613215305224}{25742413727459} a^{8} + \frac{12248938044617}{25742413727459} a^{7} + \frac{6494732975134}{25742413727459} a^{6} + \frac{1278237196900}{25742413727459} a^{5} + \frac{3284518077281}{25742413727459} a^{4} + \frac{4773301360327}{25742413727459} a^{3} - \frac{382036027583}{1980185671343} a^{2} - \frac{12830811706857}{25742413727459} a - \frac{127862541206}{362569207429}$

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:  \( 28815.0662558 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.D_4$ (as 16T339):

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 $C_2^4.D_4$
Character table for $C_2^4.D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.2525.1, 8.4.325188753125.3, 8.0.643938125.1, 8.4.3219690625.3

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$101$101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$