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Properties

Label 16.0.38385422573...5625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 11^{6}\cdot 31^{6}$
Root discriminant $29.79$
Ramified primes $5, 11, 31$
Class number $16$ (GRH)
Class group $[4, 4]$ (GRH)
Galois group $(C_2\times C_8):C_2^2$ (as 16T75)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![625, 625, 0, -750, 700, 375, 950, -280, -19, -262, 92, 45, 56, 0, -2, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 2*x^14 + 56*x^12 + 45*x^11 + 92*x^10 - 262*x^9 - 19*x^8 - 280*x^7 + 950*x^6 + 375*x^5 + 700*x^4 - 750*x^3 + 625*x + 625)
 
gp: K = bnfinit(x^16 - x^15 - 2*x^14 + 56*x^12 + 45*x^11 + 92*x^10 - 262*x^9 - 19*x^8 - 280*x^7 + 950*x^6 + 375*x^5 + 700*x^4 - 750*x^3 + 625*x + 625, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 2 x^{14} + 56 x^{12} + 45 x^{11} + 92 x^{10} - 262 x^{9} - 19 x^{8} - 280 x^{7} + 950 x^{6} + 375 x^{5} + 700 x^{4} - 750 x^{3} + 625 x + 625 \)

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:  \(383854225736338134765625=5^{12}\cdot 11^{6}\cdot 31^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.79$
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:  $5, 11, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5} a^{10} - \frac{1}{5} a^{9} - \frac{2}{5} a^{8} + \frac{1}{5} a^{6} + \frac{2}{5} a^{4} - \frac{2}{5} a^{3} + \frac{1}{5} a^{2}$, $\frac{1}{5} a^{11} + \frac{2}{5} a^{9} - \frac{2}{5} a^{8} + \frac{1}{5} a^{7} + \frac{1}{5} a^{6} + \frac{2}{5} a^{5} - \frac{1}{5} a^{3} + \frac{1}{5} a^{2}$, $\frac{1}{25} a^{12} - \frac{1}{25} a^{11} - \frac{2}{25} a^{10} + \frac{6}{25} a^{8} - \frac{1}{5} a^{7} - \frac{8}{25} a^{6} - \frac{12}{25} a^{5} + \frac{6}{25} a^{4} - \frac{1}{5} a^{3}$, $\frac{1}{25} a^{13} + \frac{2}{25} a^{11} - \frac{2}{25} a^{10} - \frac{9}{25} a^{9} - \frac{9}{25} a^{8} - \frac{8}{25} a^{7} + \frac{2}{5} a^{6} + \frac{4}{25} a^{5} + \frac{1}{25} a^{4} - \frac{2}{5} a^{3} + \frac{1}{5} a^{2}$, $\frac{1}{125} a^{14} - \frac{1}{125} a^{13} - \frac{2}{125} a^{12} + \frac{6}{125} a^{10} - \frac{6}{25} a^{9} - \frac{58}{125} a^{8} - \frac{12}{125} a^{7} + \frac{56}{125} a^{6} - \frac{6}{25} a^{5} - \frac{1}{5} a^{4} - \frac{1}{5} a^{3} + \frac{1}{5} a^{2}$, $\frac{1}{1862083142105300125} a^{15} + \frac{2126393651295387}{1862083142105300125} a^{14} - \frac{489866702207709}{33856057129187275} a^{13} - \frac{767888707212181}{169280285645936375} a^{12} + \frac{185080449606973361}{1862083142105300125} a^{11} + \frac{13999163848520238}{1862083142105300125} a^{10} - \frac{366351530792921103}{1862083142105300125} a^{9} - \frac{58183039974703901}{169280285645936375} a^{8} + \frac{121956601372923883}{372416628421060025} a^{7} - \frac{12061773010704482}{1862083142105300125} a^{6} + \frac{115362439251617139}{372416628421060025} a^{5} - \frac{60653088146584454}{372416628421060025} a^{4} + \frac{192084499625175}{1354242285167491} a^{3} - \frac{15896462715777652}{74483325684212005} a^{2} + \frac{7431583367836517}{14896665136842401} a + \frac{1238294292175746}{14896665136842401}$

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

Class group and class number

$C_{4}\times C_{4}$, which has order $16$ (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:  \( -\frac{5635216416516697}{1862083142105300125} a^{15} + \frac{14531578683161717}{1862083142105300125} a^{14} + \frac{279125905332754}{169280285645936375} a^{13} - \frac{659730864052576}{33856057129187275} a^{12} - \frac{272150053453908827}{1862083142105300125} a^{11} + \frac{50788243997303499}{372416628421060025} a^{10} - \frac{195248810289992229}{1862083142105300125} a^{9} + \frac{134576495434908219}{169280285645936375} a^{8} - \frac{1427941773432126807}{1862083142105300125} a^{7} + \frac{3810035005462337}{14896665136842401} a^{6} - \frac{710716803288527388}{372416628421060025} a^{5} + \frac{356994902084516186}{372416628421060025} a^{4} + \frac{2990280177084901}{6771211425837455} a^{3} + \frac{8597342956954119}{14896665136842401} a^{2} - \frac{16157198698796904}{14896665136842401} a + \frac{9305035746324178}{14896665136842401} \) (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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 99956.410024 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times C_8):C_2^2$ (as 16T75):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 22 conjugacy class representatives for $(C_2\times C_8):C_2^2$
Character table for $(C_2\times C_8):C_2^2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 4.4.8525.1, 4.0.42625.1, 8.4.619559703125.1, 8.4.24782388125.2, 8.0.1816890625.5

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
Arithmetically equvalently 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 ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
31Data not computed