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Properties

Label 16.0.13573982477...7041.2
Degree $16$
Signature $[0, 8]$
Discriminant $13^{12}\cdot 17^{12}$
Root discriminant $57.32$
Ramified primes $13, 17$
Class number $40$ (GRH)
Class group $[2, 2, 10]$ (GRH)
Galois group $C_4^2$ (as 16T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![23699, -36777, 45006, 27781, -27083, -17091, 8065, 9181, 3890, -1671, -389, 178, 176, -28, -19, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 19*x^14 - 28*x^13 + 176*x^12 + 178*x^11 - 389*x^10 - 1671*x^9 + 3890*x^8 + 9181*x^7 + 8065*x^6 - 17091*x^5 - 27083*x^4 + 27781*x^3 + 45006*x^2 - 36777*x + 23699)
 
gp: K = bnfinit(x^16 - 19*x^14 - 28*x^13 + 176*x^12 + 178*x^11 - 389*x^10 - 1671*x^9 + 3890*x^8 + 9181*x^7 + 8065*x^6 - 17091*x^5 - 27083*x^4 + 27781*x^3 + 45006*x^2 - 36777*x + 23699, 1)
 

Normalized defining polynomial

\( x^{16} - 19 x^{14} - 28 x^{13} + 176 x^{12} + 178 x^{11} - 389 x^{10} - 1671 x^{9} + 3890 x^{8} + 9181 x^{7} + 8065 x^{6} - 17091 x^{5} - 27083 x^{4} + 27781 x^{3} + 45006 x^{2} - 36777 x + 23699 \)

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:  \(13573982477229290545823357041=13^{12}\cdot 17^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $57.32$
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:  $13, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(221=13\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{221}(64,·)$, $\chi_{221}(1,·)$, $\chi_{221}(135,·)$, $\chi_{221}(200,·)$, $\chi_{221}(203,·)$, $\chi_{221}(18,·)$, $\chi_{221}(21,·)$, $\chi_{221}(86,·)$, $\chi_{221}(220,·)$, $\chi_{221}(157,·)$, $\chi_{221}(38,·)$, $\chi_{221}(103,·)$, $\chi_{221}(174,·)$, $\chi_{221}(47,·)$, $\chi_{221}(118,·)$, $\chi_{221}(183,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{12} a^{12} - \frac{1}{4} a^{11} - \frac{1}{12} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} + \frac{1}{4} a^{5} - \frac{1}{3} a^{4} + \frac{1}{12} a^{3} - \frac{5}{12} a^{2} + \frac{1}{6} a - \frac{5}{12}$, $\frac{1}{12} a^{13} + \frac{1}{6} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} + \frac{1}{4} a^{6} - \frac{1}{12} a^{5} - \frac{5}{12} a^{4} - \frac{1}{6} a^{3} - \frac{1}{12} a^{2} - \frac{5}{12} a - \frac{1}{4}$, $\frac{1}{60} a^{14} - \frac{1}{60} a^{12} - \frac{1}{20} a^{11} - \frac{1}{5} a^{10} - \frac{1}{10} a^{9} - \frac{1}{4} a^{8} - \frac{2}{5} a^{7} + \frac{1}{12} a^{6} - \frac{13}{30} a^{5} + \frac{1}{6} a^{4} - \frac{1}{15} a^{3} + \frac{11}{30} a^{2} - \frac{7}{20} a + \frac{9}{20}$, $\frac{1}{26445054850928824703609106573699660} a^{15} - \frac{12033731247818544938432565780677}{6611263712732206175902276643424915} a^{14} - \frac{578984363891092865434221189831311}{26445054850928824703609106573699660} a^{13} - \frac{166278955316223304672646006333479}{5289010970185764940721821314739932} a^{12} + \frac{1006975226884529557723064366773991}{13222527425464412351804553286849830} a^{11} + \frac{127242014167622436549589034675486}{1322252742546441235180455328684983} a^{10} - \frac{1102922035927074369962574962273049}{8815018283642941567869702191233220} a^{9} - \frac{249753009024504974162381575648812}{2203754570910735391967425547808305} a^{8} + \frac{3591735964157441260938670282435817}{26445054850928824703609106573699660} a^{7} + \frac{1332943668542946119392405269174209}{4407509141821470783934851095616610} a^{6} - \frac{351433356856523501470205720511586}{2203754570910735391967425547808305} a^{5} + \frac{95904379929398505967799015814173}{2203754570910735391967425547808305} a^{4} + \frac{717560850954022222634370091551181}{6611263712732206175902276643424915} a^{3} + \frac{5353772528275138509838169991933703}{26445054850928824703609106573699660} a^{2} - \frac{125102008977354070098197991519297}{1763003656728588313573940438246644} a + \frac{521093644271396825469634288993676}{6611263712732206175902276643424915}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{10}$, which has order $40$ (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:  \( 534329.701793 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4^2$ (as 16T4):

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 $C_4^2$
Character table for $C_4^2$

Intermediate fields

\(\Q(\sqrt{221}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{13}) \), 4.0.10793861.1, \(\Q(\sqrt{13}, \sqrt{17})\), 4.0.10793861.2, 4.4.4913.1, 4.4.830297.1, 4.0.2197.1, 4.0.634933.1, 8.0.116507435287321.1, 8.8.689393108209.1, 8.0.403139914489.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$13$13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
$17$17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$