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Properties

Label 16.16.1035913206...4289.2
Degree $16$
Signature $[16, 0]$
Discriminant $37^{14}\cdot 101^{14}$
Root discriminant $1336.47$
Ramified primes $37, 101$
Class number $20$ (GRH)
Class group $[20]$ (GRH)
Galois group $C_2^3.C_4$ (as 16T36)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-25448748825979, 78330056885859, -94914904681696, 55925689327219, -14975139902151, 400115166704, 638807569630, -98875841022, -4621996848, 1864600258, -58356778, -10799728, 655405, 16523, -1520, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 - 1520*x^14 + 16523*x^13 + 655405*x^12 - 10799728*x^11 - 58356778*x^10 + 1864600258*x^9 - 4621996848*x^8 - 98875841022*x^7 + 638807569630*x^6 + 400115166704*x^5 - 14975139902151*x^4 + 55925689327219*x^3 - 94914904681696*x^2 + 78330056885859*x - 25448748825979)
 
gp: K = bnfinit(x^16 - 5*x^15 - 1520*x^14 + 16523*x^13 + 655405*x^12 - 10799728*x^11 - 58356778*x^10 + 1864600258*x^9 - 4621996848*x^8 - 98875841022*x^7 + 638807569630*x^6 + 400115166704*x^5 - 14975139902151*x^4 + 55925689327219*x^3 - 94914904681696*x^2 + 78330056885859*x - 25448748825979, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} - 1520 x^{14} + 16523 x^{13} + 655405 x^{12} - 10799728 x^{11} - 58356778 x^{10} + 1864600258 x^{9} - 4621996848 x^{8} - 98875841022 x^{7} + 638807569630 x^{6} + 400115166704 x^{5} - 14975139902151 x^{4} + 55925689327219 x^{3} - 94914904681696 x^{2} + 78330056885859 x - 25448748825979 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[16, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(103591320678630890586725126366901805607295040854289=37^{14}\cdot 101^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1336.47$
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:  $37, 101$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{6} - \frac{1}{4}$, $\frac{1}{4} a^{7} - \frac{1}{4} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{16} a^{9} - \frac{1}{8} a^{8} - \frac{1}{8} a^{7} + \frac{1}{16} a^{6} + \frac{3}{16} a^{3} - \frac{3}{8} a^{2} - \frac{3}{8} a + \frac{3}{16}$, $\frac{1}{48} a^{10} + \frac{1}{48} a^{9} + \frac{1}{12} a^{8} + \frac{1}{16} a^{7} - \frac{1}{48} a^{6} + \frac{1}{16} a^{4} - \frac{5}{48} a^{3} - \frac{5}{12} a^{2} + \frac{3}{16} a - \frac{11}{48}$, $\frac{1}{48} a^{11} + \frac{5}{48} a^{8} + \frac{1}{24} a^{7} - \frac{1}{24} a^{6} + \frac{1}{16} a^{5} - \frac{1}{6} a^{4} - \frac{1}{48} a^{2} - \frac{1}{24} a - \frac{11}{24}$, $\frac{1}{288} a^{12} + \frac{1}{144} a^{10} - \frac{1}{36} a^{9} + \frac{7}{72} a^{8} + \frac{5}{144} a^{7} + \frac{11}{144} a^{6} - \frac{1}{9} a^{5} + \frac{3}{16} a^{4} + \frac{5}{36} a^{3} - \frac{1}{24} a^{2} - \frac{41}{144} a - \frac{7}{288}$, $\frac{1}{288} a^{13} + \frac{1}{144} a^{11} - \frac{1}{144} a^{10} - \frac{1}{144} a^{9} + \frac{17}{144} a^{8} - \frac{1}{9} a^{7} - \frac{1}{144} a^{6} + \frac{3}{16} a^{5} + \frac{29}{144} a^{4} - \frac{1}{48} a^{3} + \frac{43}{144} a^{2} + \frac{119}{288} a - \frac{17}{48}$, $\frac{1}{864} a^{14} - \frac{1}{864} a^{13} + \frac{1}{864} a^{12} + \frac{1}{108} a^{11} - \frac{1}{432} a^{10} + \frac{13}{432} a^{9} - \frac{35}{432} a^{8} - \frac{2}{27} a^{7} - \frac{5}{54} a^{6} - \frac{1}{4} a^{5} - \frac{107}{432} a^{4} - \frac{73}{432} a^{3} - \frac{25}{288} a^{2} - \frac{199}{864} a + \frac{295}{864}$, $\frac{1}{10237709505131647810743981385767909290190969666391247271833272835057172554027136} a^{15} + \frac{1949856432531817892964324301285632825811153152171019840189523538628994312093}{5118854752565823905371990692883954645095484833195623635916636417528586277013568} a^{14} - \frac{2538739294470433604301289956792693267965384965406222074593510676669330645889}{5118854752565823905371990692883954645095484833195623635916636417528586277013568} a^{13} - \frac{3163973001919171816997762477688287682070908348900397663112355368310850497071}{10237709505131647810743981385767909290190969666391247271833272835057172554027136} a^{12} - \frac{3684270970368730472633303798837854181541211846511745793684965660960979808293}{639856844070727988171498836610494330636935604149452954489579552191073284626696} a^{11} + \frac{1835428421783827797541558694450327927992032712019293175564755883156671636559}{319928422035363994085749418305247165318467802074726477244789776095536642313348} a^{10} - \frac{769699807314704066569178734802434146507779821062334999583076301517694723415}{189587213057993477976740396032739060929462401229467542070986533982540232481984} a^{9} - \frac{1808842392897718166031960253253972833312456570045657516283982082915905559583}{853142458760970650895331782147325774182580805532603939319439402921431046168928} a^{8} - \frac{121579822107701146143017102814835750223835454439953163913160084365439254984187}{2559427376282911952685995346441977322547742416597811817958318208764293138506784} a^{7} - \frac{434438001560330812195078384653571207808083786060254873801876469554570814941773}{5118854752565823905371990692883954645095484833195623635916636417528586277013568} a^{6} + \frac{153827878618344605724127890928832049025748191908476949908677720608088890961829}{639856844070727988171498836610494330636935604149452954489579552191073284626696} a^{5} + \frac{50319432307643413775582167348893423492837662526799941862180554916544345810437}{319928422035363994085749418305247165318467802074726477244789776095536642313348} a^{4} - \frac{1274339422853350331179027798253535921461819008746265520145144772808581187615767}{10237709505131647810743981385767909290190969666391247271833272835057172554027136} a^{3} + \frac{1068371852660561768519866410160903665412653529475612445204889004573602414608877}{5118854752565823905371990692883954645095484833195623635916636417528586277013568} a^{2} + \frac{1790935625948976758597633398251183135808632102215335609239788088030032389261927}{5118854752565823905371990692883954645095484833195623635916636417528586277013568} a + \frac{4593568305359604755181825131634164063427229471935277026913759829264389751925977}{10237709505131647810743981385767909290190969666391247271833272835057172554027136}$

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

Class group and class number

$C_{20}$, which has order $20$ (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:  $15$
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:  \( 19385888029400000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_4$ (as 16T36):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $C_2^3.C_4$
Character table for $C_2^3.C_4$

Intermediate fields

\(\Q(\sqrt{101}) \), \(\Q(\sqrt{3737}) \), \(\Q(\sqrt{37}) \), 4.4.52187836553.2, 4.4.52187836553.1, \(\Q(\sqrt{37}, \sqrt{101})\), 8.8.10177982151616836598800233.2 x2, 8.8.10177982151616836598800233.1 x2, 8.8.2723570284082642921809.3

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

Sibling fields

Galois closure: data not computed
Degree 8 siblings: data not computed
Degree 16 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$37$37.8.7.2$x^{8} - 148$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
37.8.7.2$x^{8} - 148$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$101$101.8.7.1$x^{8} - 101$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
101.8.7.1$x^{8} - 101$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$