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Properties

Label 16.0.36683429626...9169.4
Degree $16$
Signature $[0, 8]$
Discriminant $53^{14}\cdot 149^{14}$
Root discriminant $2572.05$
Ramified primes $53, 149$
Class number $17388982800$ (GRH)
Class group $[3, 6, 630, 1533420]$ (GRH)
Galois group $C_8.C_4$ (as 16T49)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![78442230270956678147, -20096776454760585706, 2220052595158974387, -111896064809531971, 4297633387795179, -55659562317467, 4371887866584, 1240781089975, -1389972933, 7541456698, 211899620, 12895156, 505614, 17359, 757, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 + 757*x^14 + 17359*x^13 + 505614*x^12 + 12895156*x^11 + 211899620*x^10 + 7541456698*x^9 - 1389972933*x^8 + 1240781089975*x^7 + 4371887866584*x^6 - 55659562317467*x^5 + 4297633387795179*x^4 - 111896064809531971*x^3 + 2220052595158974387*x^2 - 20096776454760585706*x + 78442230270956678147)
 
gp: K = bnfinit(x^16 - 7*x^15 + 757*x^14 + 17359*x^13 + 505614*x^12 + 12895156*x^11 + 211899620*x^10 + 7541456698*x^9 - 1389972933*x^8 + 1240781089975*x^7 + 4371887866584*x^6 - 55659562317467*x^5 + 4297633387795179*x^4 - 111896064809531971*x^3 + 2220052595158974387*x^2 - 20096776454760585706*x + 78442230270956678147, 1)
 

Normalized defining polynomial

\( x^{16} - 7 x^{15} + 757 x^{14} + 17359 x^{13} + 505614 x^{12} + 12895156 x^{11} + 211899620 x^{10} + 7541456698 x^{9} - 1389972933 x^{8} + 1240781089975 x^{7} + 4371887866584 x^{6} - 55659562317467 x^{5} + 4297633387795179 x^{4} - 111896064809531971 x^{3} + 2220052595158974387 x^{2} - 20096776454760585706 x + 78442230270956678147 \)

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:  \(3668342962637889127310596327197122046055622675946369169=53^{14}\cdot 149^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $2572.05$
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:  $53, 149$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7} a^{6} + \frac{3}{7} a^{5} + \frac{2}{7} a^{4} - \frac{1}{7} a^{3} - \frac{3}{7} a^{2} - \frac{2}{7} a$, $\frac{1}{7} a^{7} - \frac{1}{7} a$, $\frac{1}{7} a^{8} - \frac{1}{7} a^{2}$, $\frac{1}{7} a^{9} - \frac{1}{7} a^{3}$, $\frac{1}{7} a^{10} - \frac{1}{7} a^{4}$, $\frac{1}{833} a^{11} + \frac{1}{833} a^{10} - \frac{13}{833} a^{9} + \frac{26}{833} a^{8} + \frac{44}{833} a^{7} + \frac{55}{833} a^{6} - \frac{333}{833} a^{5} - \frac{353}{833} a^{4} - \frac{5}{119} a^{3} - \frac{10}{49} a^{2} - \frac{15}{119} a$, $\frac{1}{833} a^{12} - \frac{2}{119} a^{10} + \frac{39}{833} a^{9} + \frac{18}{833} a^{8} + \frac{11}{833} a^{7} - \frac{31}{833} a^{6} + \frac{218}{833} a^{5} + \frac{199}{833} a^{4} + \frac{341}{833} a^{3} - \frac{173}{833} a^{2} + \frac{32}{119} a$, $\frac{1}{833} a^{13} + \frac{53}{833} a^{10} - \frac{45}{833} a^{9} + \frac{18}{833} a^{8} - \frac{10}{833} a^{7} + \frac{36}{833} a^{6} + \frac{178}{833} a^{5} + \frac{159}{833} a^{4} + \frac{10}{49} a^{3} + \frac{32}{119} a^{2} + \frac{4}{17} a$, $\frac{1}{5831} a^{14} + \frac{3}{5831} a^{13} - \frac{3}{5831} a^{11} - \frac{180}{5831} a^{10} + \frac{135}{5831} a^{9} - \frac{103}{5831} a^{8} + \frac{279}{5831} a^{7} - \frac{414}{5831} a^{6} - \frac{144}{833} a^{5} - \frac{410}{5831} a^{4} - \frac{876}{5831} a^{3} + \frac{158}{833} a^{2} + \frac{30}{119} a$, $\frac{1}{939518988805513089554143334367169978307045020662891017779694576599553212330855930231436901885107090124046825168460458613} a^{15} + \frac{26504632728990958213778770977118608985149221787724472412139807635491059708227828728760592756816280866727868685775950}{939518988805513089554143334367169978307045020662891017779694576599553212330855930231436901885107090124046825168460458613} a^{14} - \frac{133219074427858018449092785895938157150396232735582956958341664696662299302548571308848215645335215240723314095019781}{939518988805513089554143334367169978307045020662891017779694576599553212330855930231436901885107090124046825168460458613} a^{13} + \frac{29974455589210151896549603215467504108407754501163871312551421857757072275771560705252943476498171650048547375394516}{939518988805513089554143334367169978307045020662891017779694576599553212330855930231436901885107090124046825168460458613} a^{12} - \frac{314252820448416006940252477107960840653474912047571312588080716793288260579478614506179856486232170292822258563666775}{939518988805513089554143334367169978307045020662891017779694576599553212330855930231436901885107090124046825168460458613} a^{11} + \frac{34874108047023313315964090763645195324224675926263027138627162860390507588504995350458616986390026673155627198214703547}{939518988805513089554143334367169978307045020662891017779694576599553212330855930231436901885107090124046825168460458613} a^{10} - \frac{47834948392582092199626110994755940791767838326729082342201854103804989940947136750968603680395805115028130041666788050}{939518988805513089554143334367169978307045020662891017779694576599553212330855930231436901885107090124046825168460458613} a^{9} - \frac{53463250658061128596346254154852482704533219058174433334385868707050216898365221139382941715157020185216238886647879472}{939518988805513089554143334367169978307045020662891017779694576599553212330855930231436901885107090124046825168460458613} a^{8} - \frac{54911523388324548389919022494738974341343802413923075443424977089909246727858158760986049161969010936938740684281407122}{939518988805513089554143334367169978307045020662891017779694576599553212330855930231436901885107090124046825168460458613} a^{7} + \frac{38990742152051479379387255212375295428374706995452321426537145006759245188880218793245657106554675418433705138499109014}{939518988805513089554143334367169978307045020662891017779694576599553212330855930231436901885107090124046825168460458613} a^{6} + \frac{41196519430045382432935633034959664064443022495216464210131365458694630831298007371119890866957395917634222002388038352}{939518988805513089554143334367169978307045020662891017779694576599553212330855930231436901885107090124046825168460458613} a^{5} + \frac{286062496389824047521580277437999261786639523994006630059007251840650448378939024951811337314618126227958017149490900770}{939518988805513089554143334367169978307045020662891017779694576599553212330855930231436901885107090124046825168460458613} a^{4} - \frac{305242083137421434404596505898369580189119175683166332213999469976794627388227968905047654383879510496087367561340857659}{939518988805513089554143334367169978307045020662891017779694576599553212330855930231436901885107090124046825168460458613} a^{3} - \frac{47029020886995959464451422064221197244333292218882993841536503022187300430928609229716606146643071130512736280291236965}{134216998400787584222020476338167139758149288666127288254242082371364744618693704318776700269301012874863832166922922659} a^{2} + \frac{818704910770379387640961800211619202870588274234989288846974970922471564651289253916097896069782023170150369237282865}{19173856914398226317431496619738162822592755523732469750606011767337820659813386331253814324185858982123404595274703237} a + \frac{1534973080443655139021517787944082750865938645712957857923217997687497654699643957030275583846279532649706049728408}{3428188255747939624071427967054919152975640179462268862972646480839946479494615828938640143784348110517326049575309}$

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

Class group and class number

$C_{3}\times C_{6}\times C_{630}\times C_{1533420}$, which has order $17388982800$ (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:  \( 227392032558 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_8.C_4$ (as 16T49):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_8.C_4$
Character table for $C_8.C_4$

Intermediate fields

\(\Q(\sqrt{53}) \), \(\Q(\sqrt{7897}) \), \(\Q(\sqrt{149}) \), \(\Q(\sqrt{53}, \sqrt{149})\), 8.8.242534110929108256632529.1

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

Sibling fields

Galois closure: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ R ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$53$53.8.7.2$x^{8} - 212$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
53.8.7.2$x^{8} - 212$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$149$149.8.7.2$x^{8} - 596$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
149.8.7.2$x^{8} - 596$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$