Properties

Label 16.0.612...689.1
Degree $16$
Signature $[0, 8]$
Discriminant $6.123\times 10^{39}$
Root discriminant $306.68$
Ramified primes $13, 41$
Class number $45284$ (GRH)
Class group $[45284]$ (GRH)
Galois group $C_{16} : C_2$ (as 16T22)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 55*x^14 + 419*x^13 - 877*x^12 + 13350*x^11 + 109599*x^10 + 897435*x^9 + 5128604*x^8 + 9987966*x^7 + 61086386*x^6 + 217020565*x^5 + 407235395*x^4 + 860961746*x^3 + 4064864216*x^2 + 670732880*x + 11844163600)
 
gp: K = bnfinit(x^16 - 7*x^15 + 55*x^14 + 419*x^13 - 877*x^12 + 13350*x^11 + 109599*x^10 + 897435*x^9 + 5128604*x^8 + 9987966*x^7 + 61086386*x^6 + 217020565*x^5 + 407235395*x^4 + 860961746*x^3 + 4064864216*x^2 + 670732880*x + 11844163600, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![11844163600, 670732880, 4064864216, 860961746, 407235395, 217020565, 61086386, 9987966, 5128604, 897435, 109599, 13350, -877, 419, 55, -7, 1]);
 

\( x^{16} - 7 x^{15} + 55 x^{14} + 419 x^{13} - 877 x^{12} + 13350 x^{11} + 109599 x^{10} + 897435 x^{9} + 5128604 x^{8} + 9987966 x^{7} + 61086386 x^{6} + 217020565 x^{5} + 407235395 x^{4} + 860961746 x^{3} + 4064864216 x^{2} + 670732880 x + 11844163600 \)

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

Invariants

Degree:  $16$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 8]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(6123007382888435990757129497254763904689\)\(\medspace = 13^{14}\cdot 41^{15}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $306.68$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $13, 41$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $8$
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{5} a^{7} - \frac{2}{5} a^{6} - \frac{2}{5} a^{5} - \frac{2}{5} a^{3} - \frac{2}{5} a^{2} - \frac{1}{5} a$, $\frac{1}{5} a^{8} - \frac{1}{5} a^{6} + \frac{1}{5} a^{5} - \frac{2}{5} a^{4} - \frac{1}{5} a^{3} - \frac{2}{5} a$, $\frac{1}{50} a^{9} - \frac{3}{50} a^{8} + \frac{3}{50} a^{7} - \frac{12}{25} a^{6} - \frac{23}{50} a^{5} + \frac{2}{5} a^{2} + \frac{17}{50} a - \frac{2}{5}$, $\frac{1}{50} a^{10} + \frac{2}{25} a^{8} - \frac{1}{10} a^{7} - \frac{1}{2} a^{6} + \frac{21}{50} a^{5} - \frac{2}{5} a^{4} - \frac{1}{5} a^{3} + \frac{7}{50} a^{2} + \frac{1}{50} a - \frac{1}{5}$, $\frac{1}{50} a^{11} - \frac{3}{50} a^{8} + \frac{3}{50} a^{7} - \frac{3}{50} a^{6} - \frac{9}{25} a^{5} + \frac{1}{5} a^{4} - \frac{13}{50} a^{3} - \frac{9}{50} a^{2} + \frac{1}{25} a - \frac{2}{5}$, $\frac{1}{250} a^{12} + \frac{1}{250} a^{11} - \frac{1}{250} a^{10} - \frac{1}{250} a^{9} + \frac{21}{250} a^{7} - \frac{62}{125} a^{6} - \frac{17}{50} a^{5} - \frac{53}{250} a^{4} - \frac{46}{125} a^{3} + \frac{28}{125} a^{2} + \frac{17}{50} a + \frac{1}{5}$, $\frac{1}{1250} a^{13} - \frac{1}{1250} a^{12} + \frac{6}{625} a^{11} + \frac{11}{1250} a^{10} + \frac{6}{625} a^{9} - \frac{57}{625} a^{8} + \frac{9}{1250} a^{7} + \frac{214}{625} a^{6} - \frac{73}{1250} a^{5} + \frac{207}{625} a^{4} + \frac{99}{250} a^{3} + \frac{29}{625} a^{2} + \frac{39}{125} a + \frac{6}{25}$, $\frac{1}{912500} a^{14} - \frac{33}{912500} a^{13} + \frac{119}{912500} a^{12} + \frac{5277}{912500} a^{11} + \frac{937}{182500} a^{10} + \frac{4301}{456250} a^{9} - \frac{37943}{912500} a^{8} + \frac{14543}{182500} a^{7} - \frac{171497}{456250} a^{6} - \frac{5353}{18250} a^{5} + \frac{73443}{228125} a^{4} - \frac{426657}{912500} a^{3} - \frac{163991}{912500} a^{2} - \frac{13579}{45625} a + \frac{169}{9125}$, $\frac{1}{18920602342436335876673264860022595479198975647263875000} a^{15} + \frac{337974786550047328660189940828102504558037514169}{3784120468487267175334652972004519095839795129452775000} a^{14} + \frac{772979987706971919755462102206762197227722991686039}{3784120468487267175334652972004519095839795129452775000} a^{13} - \frac{33534520128803473404472395515581954928174723216992041}{18920602342436335876673264860022595479198975647263875000} a^{12} - \frac{22364567444449126079098157780020458456080171916141509}{18920602342436335876673264860022595479198975647263875000} a^{11} + \frac{796530882761446087278892916464755634515166774994817}{411317442226876866884201410000491206069542948853562500} a^{10} + \frac{163335165917243853886531408943345369944173686782009963}{18920602342436335876673264860022595479198975647263875000} a^{9} - \frac{856411599262474529215294701216349386817552999993277689}{18920602342436335876673264860022595479198975647263875000} a^{8} - \frac{355464812864469632696184073346510399595635387265855131}{4730150585609083969168316215005648869799743911815968750} a^{7} + \frac{189551960647319152360486151176333835128887030539462533}{411317442226876866884201410000491206069542948853562500} a^{6} - \frac{570067892365354352325868035038736528267385557019105039}{9460301171218167938336632430011297739599487823631937500} a^{5} + \frac{2366551493087592986073797427644325972874758882187660109}{18920602342436335876673264860022595479198975647263875000} a^{4} - \frac{3865293186226563354556764240150005177465006108634657837}{18920602342436335876673264860022595479198975647263875000} a^{3} - \frac{400433048887833340235835397555030379971766978705595889}{9460301171218167938336632430011297739599487823631937500} a^{2} - \frac{8922583487636862323259872551012912689797223180421277}{20565872111343843344210070500024560303477147442678125} a + \frac{32016085701930728752074952077610358641243440449488841}{94603011712181679383366324300112977395994878236319375}$

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

Class group and class number

$C_{45284}$, which has order $45284$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $7$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 1702357630.14 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{8}\cdot 1702357630.14 \cdot 45284}{2\sqrt{6123007382888435990757129497254763904689}}\approx 1.19652600803$ (assuming GRH)

Galois group

$OD_{32}$ (as 16T22):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_{16} : C_2$
Character table for $C_{16} : C_2$

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.11647649.1, 8.8.940041681957275729.1

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

Sibling fields

Galois closure: Deg 32

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ $16$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{8}$ $16$ $16$ R $16$ $16$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ $16$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
13Data not computed
41Data not computed