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Properties

Label 16.0.43153395314...0625.4
Degree $16$
Signature $[0, 8]$
Discriminant $3^{8}\cdot 5^{8}\cdot 17^{14}$
Root discriminant $46.20$
Ramified primes $3, 5, 17$
Class number $48$ (GRH)
Class group $[48]$ (GRH)
Galois group $C_8\times C_2$ (as 16T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![41719, -63579, 25043, -11794, 27353, -5124, 25490, 6399, 4962, 1455, 1146, 24, 187, -4, 19, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 19*x^14 - 4*x^13 + 187*x^12 + 24*x^11 + 1146*x^10 + 1455*x^9 + 4962*x^8 + 6399*x^7 + 25490*x^6 - 5124*x^5 + 27353*x^4 - 11794*x^3 + 25043*x^2 - 63579*x + 41719)
 
gp: K = bnfinit(x^16 - x^15 + 19*x^14 - 4*x^13 + 187*x^12 + 24*x^11 + 1146*x^10 + 1455*x^9 + 4962*x^8 + 6399*x^7 + 25490*x^6 - 5124*x^5 + 27353*x^4 - 11794*x^3 + 25043*x^2 - 63579*x + 41719, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 19 x^{14} - 4 x^{13} + 187 x^{12} + 24 x^{11} + 1146 x^{10} + 1455 x^{9} + 4962 x^{8} + 6399 x^{7} + 25490 x^{6} - 5124 x^{5} + 27353 x^{4} - 11794 x^{3} + 25043 x^{2} - 63579 x + 41719 \)

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:  \(431533953146964646550390625=3^{8}\cdot 5^{8}\cdot 17^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $46.20$
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:  $3, 5, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(255=3\cdot 5\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{255}(1,·)$, $\chi_{255}(206,·)$, $\chi_{255}(16,·)$, $\chi_{255}(19,·)$, $\chi_{255}(149,·)$, $\chi_{255}(89,·)$, $\chi_{255}(26,·)$, $\chi_{255}(94,·)$, $\chi_{255}(161,·)$, $\chi_{255}(229,·)$, $\chi_{255}(166,·)$, $\chi_{255}(106,·)$, $\chi_{255}(236,·)$, $\chi_{255}(239,·)$, $\chi_{255}(49,·)$, $\chi_{255}(254,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{94} a^{12} + \frac{17}{94} a^{10} + \frac{3}{47} a^{9} + \frac{17}{47} a^{8} + \frac{21}{94} a^{7} - \frac{17}{47} a^{6} + \frac{15}{94} a^{5} - \frac{5}{47} a^{4} + \frac{12}{47} a^{3} - \frac{25}{94} a^{2} - \frac{23}{47} a - \frac{43}{94}$, $\frac{1}{94} a^{13} + \frac{17}{94} a^{11} + \frac{3}{47} a^{10} + \frac{17}{47} a^{9} + \frac{21}{94} a^{8} - \frac{17}{47} a^{7} + \frac{15}{94} a^{6} - \frac{5}{47} a^{5} + \frac{12}{47} a^{4} - \frac{25}{94} a^{3} - \frac{23}{47} a^{2} - \frac{43}{94} a$, $\frac{1}{188} a^{14} - \frac{1}{188} a^{13} - \frac{11}{188} a^{11} + \frac{21}{188} a^{10} - \frac{21}{188} a^{9} + \frac{25}{188} a^{8} - \frac{13}{94} a^{7} - \frac{11}{188} a^{6} - \frac{33}{188} a^{5} - \frac{67}{188} a^{4} + \frac{41}{188} a^{3} - \frac{21}{94} a^{2} + \frac{73}{188} a - \frac{21}{188}$, $\frac{1}{1146519250275775439815488304270216} a^{15} - \frac{1218950322279795510814914009437}{573259625137887719907744152135108} a^{14} + \frac{3791520069616748222532766188429}{1146519250275775439815488304270216} a^{13} - \frac{4440201844713770831554193688825}{1146519250275775439815488304270216} a^{12} + \frac{3452386687855736618042936123550}{143314906284471929976936038033777} a^{11} + \frac{51789847887790798270248308095997}{143314906284471929976936038033777} a^{10} + \frac{665459989727479698381533632203}{1623964943733392974242901280836} a^{9} - \frac{241220390638702963959834408008463}{1146519250275775439815488304270216} a^{8} + \frac{161698661180152672785521894631641}{1146519250275775439815488304270216} a^{7} + \frac{271641161496064384931815431220433}{573259625137887719907744152135108} a^{6} - \frac{2705882881148086820329472010349}{143314906284471929976936038033777} a^{5} + \frac{67449056347953714392923938084276}{143314906284471929976936038033777} a^{4} - \frac{480670699569945359248930531694279}{1146519250275775439815488304270216} a^{3} + \frac{246937650863701611483985039569253}{1146519250275775439815488304270216} a^{2} + \frac{223381111155048881721780219913081}{573259625137887719907744152135108} a + \frac{397765140971566249402957875926787}{1146519250275775439815488304270216}$

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

Class group and class number

$C_{48}$, which has order $48$ (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:  \( 81485.0410293661 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_8$ (as 16T5):

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 $C_8\times C_2$
Character table for $C_8\times C_2$

Intermediate fields

\(\Q(\sqrt{-255}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-15}, \sqrt{17})\), 4.4.4913.1, 4.0.1105425.1, 8.0.1221964430625.2, 8.8.256461670625.1, 8.0.33237432513.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
3Data not computed
5Data not computed
$17$17.8.7.1$x^{8} - 1377$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.1$x^{8} - 1377$$8$$1$$7$$C_8$$[\ ]_{8}$