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Properties

Label 16.0.20197192269...7216.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{32}\cdot 7^{8}\cdot 13^{8}$
Root discriminant $38.16$
Ramified primes $2, 7, 13$
Class number $36$ (GRH)
Class group $[3, 12]$ (GRH)
Galois group $C_2^4$ (as 16T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![22689, 65676, 40090, 21360, 65776, 25992, 13478, 11064, 7133, 504, 2186, -96, 332, -12, 28, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 28*x^14 - 12*x^13 + 332*x^12 - 96*x^11 + 2186*x^10 + 504*x^9 + 7133*x^8 + 11064*x^7 + 13478*x^6 + 25992*x^5 + 65776*x^4 + 21360*x^3 + 40090*x^2 + 65676*x + 22689)
 
gp: K = bnfinit(x^16 + 28*x^14 - 12*x^13 + 332*x^12 - 96*x^11 + 2186*x^10 + 504*x^9 + 7133*x^8 + 11064*x^7 + 13478*x^6 + 25992*x^5 + 65776*x^4 + 21360*x^3 + 40090*x^2 + 65676*x + 22689, 1)
 

Normalized defining polynomial

\( x^{16} + 28 x^{14} - 12 x^{13} + 332 x^{12} - 96 x^{11} + 2186 x^{10} + 504 x^{9} + 7133 x^{8} + 11064 x^{7} + 13478 x^{6} + 25992 x^{5} + 65776 x^{4} + 21360 x^{3} + 40090 x^{2} + 65676 x + 22689 \)

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:  \(20197192269684151435657216=2^{32}\cdot 7^{8}\cdot 13^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.16$
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:  $2, 7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(728=2^{3}\cdot 7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{728}(1,·)$, $\chi_{728}(363,·)$, $\chi_{728}(391,·)$, $\chi_{728}(209,·)$, $\chi_{728}(727,·)$, $\chi_{728}(27,·)$, $\chi_{728}(545,·)$, $\chi_{728}(547,·)$, $\chi_{728}(337,·)$, $\chi_{728}(519,·)$, $\chi_{728}(365,·)$, $\chi_{728}(573,·)$, $\chi_{728}(155,·)$, $\chi_{728}(181,·)$, $\chi_{728}(183,·)$, $\chi_{728}(701,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{6} a^{9} - \frac{1}{2} a^{5} - \frac{1}{6} a$, $\frac{1}{6} a^{10} - \frac{1}{2} a^{6} - \frac{1}{6} a^{2}$, $\frac{1}{6} a^{11} - \frac{1}{2} a^{7} - \frac{1}{6} a^{3}$, $\frac{1}{360} a^{12} + \frac{1}{45} a^{11} + \frac{1}{360} a^{10} - \frac{2}{45} a^{9} + \frac{1}{30} a^{8} - \frac{1}{2} a^{7} + \frac{1}{40} a^{6} + \frac{1}{30} a^{5} - \frac{89}{180} a^{4} + \frac{13}{90} a^{3} - \frac{97}{360} a^{2} - \frac{19}{45} a + \frac{47}{120}$, $\frac{1}{360} a^{13} - \frac{1}{120} a^{11} - \frac{1}{15} a^{10} + \frac{1}{18} a^{9} + \frac{7}{30} a^{8} - \frac{19}{40} a^{7} - \frac{1}{6} a^{6} + \frac{43}{180} a^{5} + \frac{1}{10} a^{4} + \frac{49}{120} a^{3} - \frac{4}{15} a^{2} + \frac{37}{360} a - \frac{2}{15}$, $\frac{1}{506313720} a^{14} + \frac{138659}{168771240} a^{13} + \frac{5453}{16877124} a^{12} + \frac{13497623}{168771240} a^{11} + \frac{142093}{101262744} a^{10} + \frac{2249663}{42192810} a^{9} + \frac{298307}{5114280} a^{8} + \frac{75084271}{168771240} a^{7} - \frac{978881}{29783160} a^{6} - \frac{1553461}{84385620} a^{5} - \frac{96637}{309672} a^{4} - \frac{868723}{18752360} a^{3} + \frac{2270098}{5753565} a^{2} - \frac{5965873}{33754248} a - \frac{22687127}{56257080}$, $\frac{1}{60626082259813003560} a^{15} + \frac{574446632}{688932752952420495} a^{14} + \frac{4614688379937821}{3368115681100722420} a^{13} - \frac{26339146503161213}{20208694086604334520} a^{12} - \frac{4520683373172322963}{60626082259813003560} a^{11} - \frac{93404141540678627}{60626082259813003560} a^{10} + \frac{35386625908404685}{1347246272440288968} a^{9} - \frac{345409045111625963}{5052173521651083630} a^{8} - \frac{3574223870124817337}{12125216451962600712} a^{7} - \frac{9515125513275739151}{60626082259813003560} a^{6} - \frac{85482982781260151}{1347246272440288968} a^{5} - \frac{3178326231794054557}{10104347043302167260} a^{4} - \frac{2737600954723029322}{7578260282476625445} a^{3} - \frac{8725860389049490273}{60626082259813003560} a^{2} - \frac{1034905357898171101}{2245410454067148280} a - \frac{640497090450510295}{1347246272440288968}$

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

Class group and class number

$C_{3}\times C_{12}$, which has order $36$ (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:  \( -\frac{71161925675287}{139050647384892210} a^{15} + \frac{9332074864433}{32717799384680520} a^{14} - \frac{79107383927221}{5452966564113420} a^{13} + \frac{58419410737645}{4120019181774584} a^{12} - \frac{4510714028439659}{25281935888162220} a^{11} + \frac{40981755639688909}{278101294769784420} a^{10} - \frac{9336485834430277}{7725035965827345} a^{9} + \frac{74385550270502879}{185400863179856280} a^{8} - \frac{1092217964808114151}{278101294769784420} a^{7} - \frac{497203687398421447}{139050647384892210} a^{6} - \frac{236542427473746107}{46350215794964070} a^{5} - \frac{2021153211606260363}{185400863179856280} a^{4} - \frac{7819617197124069401}{278101294769784420} a^{3} + \frac{444401040700394411}{111240517907913768} a^{2} - \frac{245150815020684863}{10300047954436460} a - \frac{633328646530832657}{30900143863309380} \) (order $8$)
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:  \( 377056.097397 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4$ (as 16T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 16
The 16 conjugacy class representatives for $C_2^4$
Character table for $C_2^4$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-182}) \), \(\Q(\sqrt{182}) \), \(\Q(\sqrt{91}) \), \(\Q(\sqrt{-91}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-13}) \), \(\Q(\sqrt{-26}) \), \(\Q(\sqrt{26}) \), \(\Q(\sqrt{-14}) \), \(\Q(\sqrt{14}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{-7}) \), \(\Q(\zeta_{8})\), \(\Q(i, \sqrt{182})\), \(\Q(i, \sqrt{91})\), \(\Q(\sqrt{-2}, \sqrt{91})\), \(\Q(\sqrt{-2}, \sqrt{-91})\), \(\Q(\sqrt{2}, \sqrt{-91})\), \(\Q(\sqrt{2}, \sqrt{91})\), \(\Q(i, \sqrt{13})\), \(\Q(i, \sqrt{26})\), \(\Q(i, \sqrt{14})\), \(\Q(i, \sqrt{7})\), \(\Q(\sqrt{-2}, \sqrt{13})\), \(\Q(\sqrt{-2}, \sqrt{-13})\), \(\Q(\sqrt{-2}, \sqrt{7})\), \(\Q(\sqrt{-2}, \sqrt{-7})\), \(\Q(\sqrt{2}, \sqrt{13})\), \(\Q(\sqrt{2}, \sqrt{-13})\), \(\Q(\sqrt{2}, \sqrt{-7})\), \(\Q(\sqrt{2}, \sqrt{7})\), \(\Q(\sqrt{13}, \sqrt{-14})\), \(\Q(\sqrt{-13}, \sqrt{14})\), \(\Q(\sqrt{7}, \sqrt{-26})\), \(\Q(\sqrt{-7}, \sqrt{26})\), \(\Q(\sqrt{13}, \sqrt{14})\), \(\Q(\sqrt{-13}, \sqrt{-14})\), \(\Q(\sqrt{-7}, \sqrt{-26})\), \(\Q(\sqrt{7}, \sqrt{26})\), \(\Q(\sqrt{7}, \sqrt{13})\), \(\Q(\sqrt{-7}, \sqrt{-13})\), \(\Q(\sqrt{-14}, \sqrt{-26})\), \(\Q(\sqrt{14}, \sqrt{26})\), \(\Q(\sqrt{-7}, \sqrt{13})\), \(\Q(\sqrt{7}, \sqrt{-13})\), \(\Q(\sqrt{14}, \sqrt{-26})\), \(\Q(\sqrt{-14}, \sqrt{26})\), 8.0.4494128644096.9, 8.0.1871773696.1, 8.0.157351936.1, 8.0.4494128644096.6, 8.0.4494128644096.8, 8.0.17555190016.1, 8.0.4494128644096.2, 8.0.4494128644096.5, 8.0.4494128644096.7, 8.0.280883040256.2, 8.0.4494128644096.1, 8.0.280883040256.1, 8.0.4494128644096.4, 8.8.4494128644096.1, 8.0.4494128644096.3

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
2.8.16.6$x^{8} + 4 x^{6} + 8 x^{2} + 4$$4$$2$$16$$C_2^3$$[2, 3]^{2}$
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$13$13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$