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Properties

Label 16.0.44148012183...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{8}\cdot 5^{14}\cdot 29^{6}\cdot 41^{6}$
Root discriminant $82.28$
Ramified primes $2, 5, 29, 41$
Class number $64$ (GRH)
Class group $[2, 2, 2, 8]$ (GRH)
Galois group $C_2^4.C_2^3.C_2$ (as 16T646)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1185921, -11444301, 35345376, -22015584, 1841738, -2245971, 4428414, -1752687, -70297, 172969, -17774, -7747, 1238, 198, -26, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 7*x^15 - 26*x^14 + 198*x^13 + 1238*x^12 - 7747*x^11 - 17774*x^10 + 172969*x^9 - 70297*x^8 - 1752687*x^7 + 4428414*x^6 - 2245971*x^5 + 1841738*x^4 - 22015584*x^3 + 35345376*x^2 - 11444301*x + 1185921)
 
gp: K = bnfinit(x^16 - 7*x^15 - 26*x^14 + 198*x^13 + 1238*x^12 - 7747*x^11 - 17774*x^10 + 172969*x^9 - 70297*x^8 - 1752687*x^7 + 4428414*x^6 - 2245971*x^5 + 1841738*x^4 - 22015584*x^3 + 35345376*x^2 - 11444301*x + 1185921, 1)
 

Normalized defining polynomial

\( x^{16} - 7 x^{15} - 26 x^{14} + 198 x^{13} + 1238 x^{12} - 7747 x^{11} - 17774 x^{10} + 172969 x^{9} - 70297 x^{8} - 1752687 x^{7} + 4428414 x^{6} - 2245971 x^{5} + 1841738 x^{4} - 22015584 x^{3} + 35345376 x^{2} - 11444301 x + 1185921 \)

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:  \(4414801218324694314062500000000=2^{8}\cdot 5^{14}\cdot 29^{6}\cdot 41^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $82.28$
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:  $2, 5, 29, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{10} + \frac{1}{4} a^{7} + \frac{1}{4} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{88} a^{12} + \frac{5}{88} a^{11} - \frac{9}{88} a^{10} - \frac{15}{88} a^{9} - \frac{9}{44} a^{8} - \frac{25}{88} a^{7} - \frac{41}{88} a^{6} - \frac{17}{88} a^{5} - \frac{5}{44} a^{4} + \frac{15}{88} a^{3} - \frac{5}{88} a^{2} + \frac{41}{88} a - \frac{1}{8}$, $\frac{1}{264} a^{13} - \frac{1}{264} a^{12} - \frac{17}{264} a^{11} - \frac{9}{88} a^{10} - \frac{2}{33} a^{9} + \frac{17}{264} a^{8} - \frac{23}{264} a^{7} - \frac{35}{264} a^{6} + \frac{23}{66} a^{5} - \frac{19}{88} a^{4} - \frac{39}{88} a^{3} - \frac{35}{88} a^{2} + \frac{29}{264} a - \frac{1}{2}$, $\frac{1}{1584} a^{14} + \frac{1}{792} a^{13} + \frac{1}{1584} a^{12} - \frac{19}{176} a^{11} - \frac{1}{72} a^{10} + \frac{91}{792} a^{9} + \frac{61}{396} a^{8} - \frac{299}{1584} a^{7} - \frac{103}{396} a^{6} + \frac{5}{11} a^{5} + \frac{5}{22} a^{4} - \frac{91}{528} a^{3} - \frac{787}{1584} a^{2} - \frac{17}{88} a - \frac{1}{16}$, $\frac{1}{1025889877084386554223191033631511858945488432} a^{15} - \frac{4034037927493992449523197518228327655707}{1025889877084386554223191033631511858945488432} a^{14} + \frac{197212518391369165549885267179589933154875}{1025889877084386554223191033631511858945488432} a^{13} + \frac{25730108470815411142675815761827190802995}{5181262005476699768803995119351069994674184} a^{12} + \frac{57267717925800021825726015923617626570351827}{1025889877084386554223191033631511858945488432} a^{11} + \frac{8277588139090534719265371221928278981790715}{512944938542193277111595516815755929472744216} a^{10} + \frac{20690599926833477047031761914982099592555491}{256472469271096638555797758407877964736372108} a^{9} - \frac{116270149262886368073606306471228384887830871}{1025889877084386554223191033631511858945488432} a^{8} + \frac{102691360789863467784010529726633894743586285}{1025889877084386554223191033631511858945488432} a^{7} + \frac{2066990619293270102882072924973901585823363}{56993882060243697456843946312861769941416024} a^{6} - \frac{8119320914051407316529003650480821517266835}{18997960686747899152281315437620589980472008} a^{5} + \frac{127674664117712129443639657921948668151373273}{341963292361462184741063677877170619648496144} a^{4} + \frac{51918893688457699705558482951949227565429733}{256472469271096638555797758407877964736372108} a^{3} + \frac{19024697073525374538719599930202001043743091}{113987764120487394913687892625723539882832048} a^{2} + \frac{1223908997469881399111809959295895453485003}{10362524010953399537607990238702139989348368} a - \frac{18037207564667219462109956511353912848245}{314015879119799985988120916324307272404496}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{8}$, which has order $64$ (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:  \( -\frac{2057955652828690545976305185033404279}{2436793057207569012406629533566536482055792} a^{15} + \frac{764221941122371538228172286477376200}{152299566075473063275414345847908530128487} a^{14} + \frac{67530626677021455517979286861355510301}{2436793057207569012406629533566536482055792} a^{13} - \frac{3353200432688579812868352939594741413}{24614071284924939519258884177439762445008} a^{12} - \frac{1475891018057059965124572131159744124661}{1218396528603784506203314766783268241027896} a^{11} + \frac{784556943669372369117162619265825175838}{152299566075473063275414345847908530128487} a^{10} + \frac{25625322062518352116568756172617002261015}{1218396528603784506203314766783268241027896} a^{9} - \frac{293065967849008260522985253322331449721927}{2436793057207569012406629533566536482055792} a^{8} - \frac{23791422108722690280729982709815349159695}{304599132150946126550828691695817060256974} a^{7} + \frac{181933671830805871484383554283327073982887}{135377392067087167355923862975918693447544} a^{6} - \frac{293564766908787681161369347765378688670853}{135377392067087167355923862975918693447544} a^{5} + \frac{4170817701263502090157146261149830309693}{812264352402523004135543177855512160685264} a^{4} - \frac{6617201468257831733397888710889371256117035}{2436793057207569012406629533566536482055792} a^{3} + \frac{1975036559560769718394844060504356554513883}{135377392067087167355923862975918693447544} a^{2} - \frac{319009291667454515950235902224831542548427}{24614071284924939519258884177439762445008} a + \frac{134249298036886926973139025135794063249}{46617559251751779392535765487575307661} \) (order $10$)
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:  \( 61162681.6721 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3.C_2$ (as 16T646):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 34 conjugacy class representatives for $C_2^4.C_2^3.C_2$
Character table for $C_2^4.C_2^3.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.148625.2, \(\Q(\zeta_{5})\), 4.0.29725.2, 8.0.22089390625.3

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

Sibling fields

Degree 16 siblings: data not computed
Degree 32 siblings: 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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$2$2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
5Data not computed
$29$29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.8.6.2$x^{8} + 145 x^{4} + 7569$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
41Data not computed