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Properties

Label 16.0.38207364016...896.22
Degree $16$
Signature $[0, 8]$
Discriminant $2^{48}\cdot 13^{12}\cdot 17^{12}$
Root discriminant $458.55$
Ramified primes $2, 13, 17$
Class number $2087747584$ (GRH)
Class group $[2, 4, 8, 104, 104, 3016]$ (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![1310484615696, 0, 131763169728, 0, 25613941120, 0, -149969952, 0, 6123000, 0, 696016, 0, 15552, 0, 168, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 168*x^14 + 15552*x^12 + 696016*x^10 + 6123000*x^8 - 149969952*x^6 + 25613941120*x^4 + 131763169728*x^2 + 1310484615696)
 
gp: K = bnfinit(x^16 + 168*x^14 + 15552*x^12 + 696016*x^10 + 6123000*x^8 - 149969952*x^6 + 25613941120*x^4 + 131763169728*x^2 + 1310484615696, 1)
 

Normalized defining polynomial

\( x^{16} + 168 x^{14} + 15552 x^{12} + 696016 x^{10} + 6123000 x^{8} - 149969952 x^{6} + 25613941120 x^{4} + 131763169728 x^{2} + 1310484615696 \)

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:  \(3820736401648967194220515025487549637328896=2^{48}\cdot 13^{12}\cdot 17^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $458.55$
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:  $2, 13, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3536=2^{4}\cdot 13\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{3536}(1,·)$, $\chi_{3536}(3141,·)$, $\chi_{3536}(1483,·)$, $\chi_{3536}(2835,·)$, $\chi_{3536}(2007,·)$, $\chi_{3536}(545,·)$, $\chi_{3536}(421,·)$, $\chi_{3536}(1191,·)$, $\chi_{3536}(3433,·)$, $\chi_{3536}(2027,·)$, $\chi_{3536}(2605,·)$, $\chi_{3536}(1903,·)$, $\chi_{3536}(3379,·)$, $\chi_{3536}(441,·)$, $\chi_{3536}(1789,·)$, $\chi_{3536}(1087,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{12} a^{4} - \frac{1}{3} a^{2} - \frac{1}{2}$, $\frac{1}{12} a^{5} + \frac{1}{6} a$, $\frac{1}{36} a^{6} + \frac{1}{36} a^{4} - \frac{1}{18} a^{2} - \frac{1}{2}$, $\frac{1}{36} a^{7} + \frac{1}{36} a^{5} - \frac{1}{18} a^{3} - \frac{1}{2} a$, $\frac{1}{25920} a^{8} - \frac{1}{72} a^{7} - \frac{13}{2160} a^{6} - \frac{1}{72} a^{5} - \frac{13}{1080} a^{4} - \frac{5}{36} a^{3} + \frac{577}{3240} a^{2} + \frac{5}{12} a + \frac{319}{720}$, $\frac{1}{25920} a^{9} - \frac{13}{2160} a^{7} - \frac{1}{72} a^{6} - \frac{13}{1080} a^{5} + \frac{1}{36} a^{4} - \frac{503}{3240} a^{3} - \frac{5}{36} a^{2} - \frac{161}{720} a - \frac{1}{2}$, $\frac{1}{25920} a^{10} - \frac{1}{72} a^{7} - \frac{7}{1080} a^{6} + \frac{1}{36} a^{5} - \frac{17}{3240} a^{4} - \frac{5}{36} a^{3} + \frac{145}{432} a^{2} - \frac{1}{2} a - \frac{23}{60}$, $\frac{1}{77760} a^{11} - \frac{1}{77760} a^{9} - \frac{61}{6480} a^{7} - \frac{1}{72} a^{6} - \frac{17}{2430} a^{5} + \frac{1}{36} a^{4} + \frac{1381}{19440} a^{3} - \frac{5}{36} a^{2} - \frac{191}{432} a - \frac{1}{2}$, $\frac{1}{38736610560} a^{12} + \frac{17893}{1936830528} a^{10} + \frac{103363}{6456101760} a^{8} - \frac{32279497}{2421038160} a^{6} - \frac{1}{24} a^{5} - \frac{391403573}{9684152640} a^{4} - \frac{1}{6} a^{3} + \frac{17314201}{53800848} a^{2} + \frac{1}{12} a + \frac{1105387}{6642080}$, $\frac{1}{38736610560} a^{13} - \frac{17537}{4842076320} a^{11} - \frac{62689}{6456101760} a^{9} + \frac{5082203}{2421038160} a^{7} - \frac{1}{72} a^{6} - \frac{207085853}{9684152640} a^{5} + \frac{1}{36} a^{4} + \frac{14666389}{201753180} a^{3} + \frac{13}{36} a^{2} + \frac{29679397}{179336160} a - \frac{1}{2}$, $\frac{1}{32676416082896613120} a^{14} - \frac{94319309}{8169104020724153280} a^{12} + \frac{112139989131719}{16338208041448306560} a^{10} - \frac{5359360822927}{816910402072415328} a^{8} - \frac{4999356819071969}{1633820804144830656} a^{6} + \frac{2046144458027953}{70423310523484080} a^{4} - \frac{1}{6} a^{3} + \frac{132135430927490197}{453839112262452960} a^{2} - \frac{1}{3} a - \frac{3693568466934617}{12606642007290360}$, $\frac{1}{1039077355020029400602880} a^{15} + \frac{2019934138429}{1039077355020029400602880} a^{13} - \frac{3299782598723293387}{519538677510014700301440} a^{11} - \frac{588051995041375741}{103907735502002940060288} a^{9} - \frac{2241892527597999741301}{259769338755007350150720} a^{7} - \frac{332835407014204144301}{8957563405345081039680} a^{5} + \frac{2136727849065739264567}{14431629930833741675040} a^{3} - \frac{1}{2} a^{2} - \frac{245527625626053952019}{1603514436759304630560} a$

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

Class group and class number

$C_{2}\times C_{4}\times C_{8}\times C_{104}\times C_{104}\times C_{3016}$, which has order $2087747584$ (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:  \( 103963078.62847929 \) (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

\(\Q(\sqrt{442}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{34}) \), 4.0.22105827328.8, \(\Q(\sqrt{13}, \sqrt{34})\), 4.0.22105827328.4, 4.4.140608.1, 4.4.10158928.1, 4.0.10061824.1, 4.0.1700448256.3, 8.0.488667601855351619584.4, 8.8.26420177435951104.3, 8.0.2891524271333441536.2

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 ${\href{/LocalNumberField/3.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
13Data not computed
$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}$