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Properties

Label 16.0.76699264189...0000.5
Degree $16$
Signature $[0, 8]$
Discriminant $2^{48}\cdot 5^{8}\cdot 17^{8}$
Root discriminant $73.76$
Ramified primes $2, 5, 17$
Class number $19968$ (GRH)
Class group $[4, 4, 4, 312]$ (GRH)
Galois group $C_4\times C_2^2$ (as 16T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![22430641, -19004192, 24807628, -12559848, 8820932, -3512472, 2002148, -835472, 372741, -125624, 42320, -10440, 2650, -448, 84, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 84*x^14 - 448*x^13 + 2650*x^12 - 10440*x^11 + 42320*x^10 - 125624*x^9 + 372741*x^8 - 835472*x^7 + 2002148*x^6 - 3512472*x^5 + 8820932*x^4 - 12559848*x^3 + 24807628*x^2 - 19004192*x + 22430641)
 
gp: K = bnfinit(x^16 - 8*x^15 + 84*x^14 - 448*x^13 + 2650*x^12 - 10440*x^11 + 42320*x^10 - 125624*x^9 + 372741*x^8 - 835472*x^7 + 2002148*x^6 - 3512472*x^5 + 8820932*x^4 - 12559848*x^3 + 24807628*x^2 - 19004192*x + 22430641, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 84 x^{14} - 448 x^{13} + 2650 x^{12} - 10440 x^{11} + 42320 x^{10} - 125624 x^{9} + 372741 x^{8} - 835472 x^{7} + 2002148 x^{6} - 3512472 x^{5} + 8820932 x^{4} - 12559848 x^{3} + 24807628 x^{2} - 19004192 x + 22430641 \)

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:  \(766992641892445428121600000000=2^{48}\cdot 5^{8}\cdot 17^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $73.76$
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:  $2, 5, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1360=2^{4}\cdot 5\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{1360}(1,·)$, $\chi_{1360}(69,·)$, $\chi_{1360}(1089,·)$, $\chi_{1360}(1291,·)$, $\chi_{1360}(271,·)$, $\chi_{1360}(339,·)$, $\chi_{1360}(341,·)$, $\chi_{1360}(409,·)$, $\chi_{1360}(1359,·)$, $\chi_{1360}(611,·)$, $\chi_{1360}(679,·)$, $\chi_{1360}(681,·)$, $\chi_{1360}(749,·)$, $\chi_{1360}(951,·)$, $\chi_{1360}(1019,·)$, $\chi_{1360}(1021,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7} a^{6} - \frac{3}{7} a^{5} + \frac{2}{7} a^{4} + \frac{1}{7} a^{3} - \frac{3}{7} a^{2} + \frac{2}{7} a + \frac{2}{7}$, $\frac{1}{7} a^{7} + \frac{1}{7} a - \frac{1}{7}$, $\frac{1}{63} a^{8} - \frac{4}{63} a^{7} - \frac{2}{63} a^{6} + \frac{20}{63} a^{5} + \frac{17}{63} a^{4} - \frac{1}{7} a^{3} - \frac{1}{3} a^{2} - \frac{2}{63} a - \frac{1}{9}$, $\frac{1}{63} a^{9} + \frac{1}{21} a^{6} - \frac{2}{63} a^{5} - \frac{22}{63} a^{4} - \frac{1}{21} a^{3} + \frac{4}{63} a^{2} - \frac{5}{21} a - \frac{1}{63}$, $\frac{1}{441} a^{10} + \frac{2}{441} a^{9} - \frac{1}{147} a^{8} - \frac{1}{21} a^{7} - \frac{17}{441} a^{6} - \frac{68}{441} a^{5} - \frac{89}{441} a^{4} - \frac{2}{441} a^{3} - \frac{178}{441} a^{2} + \frac{11}{441} a + \frac{190}{441}$, $\frac{1}{441} a^{11} - \frac{1}{441} a^{8} - \frac{31}{441} a^{7} + \frac{22}{441} a^{6} + \frac{124}{441} a^{5} - \frac{55}{441} a^{4} + \frac{61}{147} a^{3} - \frac{88}{441} a^{2} + \frac{23}{63} a + \frac{82}{441}$, $\frac{1}{71001} a^{12} - \frac{2}{23667} a^{11} - \frac{22}{71001} a^{10} + \frac{55}{23667} a^{9} - \frac{526}{71001} a^{8} + \frac{1048}{71001} a^{7} - \frac{928}{23667} a^{6} + \frac{5443}{71001} a^{5} - \frac{4663}{10143} a^{4} - \frac{13931}{71001} a^{3} - \frac{2853}{7889} a^{2} - \frac{2071}{71001} a - \frac{1354}{71001}$, $\frac{1}{5609079} a^{13} + \frac{11}{1869693} a^{12} + \frac{344}{1869693} a^{11} + \frac{706}{801297} a^{10} - \frac{16631}{5609079} a^{9} + \frac{11114}{1869693} a^{8} + \frac{5800}{243873} a^{7} + \frac{118948}{1869693} a^{6} + \frac{2681254}{5609079} a^{5} + \frac{1207282}{5609079} a^{4} + \frac{1379758}{5609079} a^{3} + \frac{86792}{1869693} a^{2} - \frac{803126}{1869693} a - \frac{2524639}{5609079}$, $\frac{1}{5482308205521} a^{14} - \frac{1}{783186886503} a^{13} + \frac{6324610}{5482308205521} a^{12} - \frac{37947569}{5482308205521} a^{11} + \frac{1478231863}{1827436068507} a^{10} - \frac{21825625396}{5482308205521} a^{9} - \frac{7592869}{7129139409} a^{8} + \frac{153891994447}{5482308205521} a^{7} + \frac{30407009872}{1827436068507} a^{6} - \frac{903535273117}{5482308205521} a^{5} - \frac{344576128136}{783186886503} a^{4} + \frac{1070632909747}{5482308205521} a^{3} - \frac{162479384279}{609145356169} a^{2} - \frac{1996910041718}{5482308205521} a - \frac{292292989}{783186886503}$, $\frac{1}{75431078599763439} a^{15} + \frac{6872}{75431078599763439} a^{14} - \frac{861653855}{75431078599763439} a^{13} + \frac{7015798807}{10775868371394777} a^{12} - \frac{2379957993848}{3279612113033193} a^{11} - \frac{12015321347150}{75431078599763439} a^{10} + \frac{14557966205652}{8381230955529271} a^{9} + \frac{18187928899481}{3591956123798259} a^{8} + \frac{56821048359871}{25143692866587813} a^{7} + \frac{4091420679618946}{75431078599763439} a^{6} + \frac{20558196078912754}{75431078599763439} a^{5} - \frac{272052450572863}{613260801624093} a^{4} + \frac{14267789314989302}{75431078599763439} a^{3} - \frac{2794370846167419}{8381230955529271} a^{2} + \frac{526314664684712}{75431078599763439} a + \frac{3252658077358330}{8381230955529271}$

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

Class group and class number

$C_{4}\times C_{4}\times C_{4}\times C_{312}$, which has order $19968$ (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:  \( 12198.951274811623 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_4$ (as 16T2):

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 $C_4\times C_2^2$
Character table for $C_4\times C_2^2$

Intermediate fields

\(\Q(\sqrt{10}) \), \(\Q(\sqrt{-85}) \), \(\Q(\sqrt{-34}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-170}) \), \(\Q(\sqrt{-17}) \), \(\Q(\sqrt{10}, \sqrt{-34})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{10}, \sqrt{-17})\), \(\Q(\sqrt{2}, \sqrt{-85})\), \(\Q(\sqrt{5}, \sqrt{-17})\), \(\Q(\sqrt{2}, \sqrt{-17})\), \(\Q(\sqrt{5}, \sqrt{-34})\), \(\Q(\zeta_{16})^+\), 4.4.51200.1, 4.0.14796800.5, 4.0.591872.5, 8.0.3421020160000.8, 8.8.2621440000.1, 8.0.218945290240000.141, 8.0.875781160960000.56, 8.0.875781160960000.36, 8.0.1401249857536.2, 8.0.875781160960000.45

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
2Data not computed
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$17$17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$