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Properties

Label 16.8.10998771473...7801.2
Degree $16$
Signature $[8, 4]$
Discriminant $37^{14}\cdot 73^{14}$
Root discriminant $1005.97$
Ramified primes $37, 73$
Class number $4$ (GRH)
Class group $[4]$ (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![550007954217, 2244417237747, 259076011275, -221952921930, -93063891411, 20777765544, -282326958, -177168789, 95292171, -5090319, -331733, 23439, -18187, 119, -29, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 29*x^14 + 119*x^13 - 18187*x^12 + 23439*x^11 - 331733*x^10 - 5090319*x^9 + 95292171*x^8 - 177168789*x^7 - 282326958*x^6 + 20777765544*x^5 - 93063891411*x^4 - 221952921930*x^3 + 259076011275*x^2 + 2244417237747*x + 550007954217)
 
gp: K = bnfinit(x^16 - 4*x^15 - 29*x^14 + 119*x^13 - 18187*x^12 + 23439*x^11 - 331733*x^10 - 5090319*x^9 + 95292171*x^8 - 177168789*x^7 - 282326958*x^6 + 20777765544*x^5 - 93063891411*x^4 - 221952921930*x^3 + 259076011275*x^2 + 2244417237747*x + 550007954217, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 29 x^{14} + 119 x^{13} - 18187 x^{12} + 23439 x^{11} - 331733 x^{10} - 5090319 x^{9} + 95292171 x^{8} - 177168789 x^{7} - 282326958 x^{6} + 20777765544 x^{5} - 93063891411 x^{4} - 221952921930 x^{3} + 259076011275 x^{2} + 2244417237747 x + 550007954217 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1099877147396767625299704166328746389549375427801=37^{14}\cdot 73^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1005.97$
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:  $37, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{9} a^{6} - \frac{1}{9} a^{4} + \frac{4}{9} a^{3} - \frac{1}{9} a^{2} - \frac{1}{3} a$, $\frac{1}{27} a^{7} - \frac{1}{27} a^{5} + \frac{4}{27} a^{4} - \frac{1}{27} a^{3} - \frac{4}{9} a^{2} - \frac{1}{3} a$, $\frac{1}{1971} a^{8} - \frac{2}{1971} a^{7} + \frac{20}{1971} a^{6} - \frac{52}{657} a^{5} + \frac{71}{657} a^{4} - \frac{466}{1971} a^{3} - \frac{11}{657} a^{2} - \frac{71}{219} a - \frac{35}{73}$, $\frac{1}{1971} a^{9} + \frac{16}{1971} a^{7} + \frac{103}{1971} a^{6} - \frac{11}{219} a^{5} - \frac{259}{1971} a^{4} - \frac{89}{1971} a^{3} - \frac{308}{657} a^{2} - \frac{101}{219} a + \frac{3}{73}$, $\frac{1}{5913} a^{10} - \frac{1}{5913} a^{9} + \frac{1}{5913} a^{8} + \frac{44}{5913} a^{7} + \frac{155}{5913} a^{6} - \frac{125}{1971} a^{5} - \frac{689}{5913} a^{4} - \frac{37}{219} a^{3} + \frac{8}{657} a^{2} - \frac{95}{219} a + \frac{28}{73}$, $\frac{1}{53217} a^{11} - \frac{4}{53217} a^{10} + \frac{7}{53217} a^{9} + \frac{2}{53217} a^{8} + \frac{587}{53217} a^{7} - \frac{218}{17739} a^{6} - \frac{8012}{53217} a^{5} + \frac{983}{5913} a^{4} - \frac{2}{219} a^{3} - \frac{223}{657} a^{2} + \frac{251}{657} a + \frac{1}{73}$, $\frac{1}{319302} a^{12} + \frac{1}{159651} a^{11} - \frac{17}{319302} a^{10} - \frac{32}{159651} a^{9} - \frac{11}{159651} a^{8} - \frac{260}{53217} a^{7} + \frac{5912}{159651} a^{6} + \frac{443}{106434} a^{5} + \frac{335}{3942} a^{4} + \frac{625}{1314} a^{3} + \frac{1493}{3942} a^{2} + \frac{176}{657} a + \frac{23}{146}$, $\frac{1}{2873718} a^{13} + \frac{1}{1436859} a^{12} - \frac{17}{2873718} a^{11} - \frac{32}{1436859} a^{10} - \frac{11}{1436859} a^{9} - \frac{98}{478953} a^{8} - \frac{12799}{1436859} a^{7} + \frac{6923}{957906} a^{6} + \frac{2843}{35478} a^{5} - \frac{421}{11826} a^{4} - \frac{15487}{35478} a^{3} + \frac{2300}{5913} a^{2} + \frac{323}{1314} a + \frac{1}{73}$, $\frac{1}{121187561778} a^{14} - \frac{5150}{60593780889} a^{13} - \frac{133157}{121187561778} a^{12} - \frac{350135}{60593780889} a^{11} + \frac{2615905}{60593780889} a^{10} - \frac{1941352}{20197926963} a^{9} - \frac{7773886}{60593780889} a^{8} - \frac{187724693}{13465284642} a^{7} - \frac{516734509}{13465284642} a^{6} - \frac{34802549}{1496142738} a^{5} + \frac{2946485}{1496142738} a^{4} + \frac{1562452}{27706347} a^{3} - \frac{20755625}{166238082} a^{2} + \frac{1165760}{9235449} a + \frac{310624}{1026161}$, $\frac{1}{10127591350082033823995260715932509181219406162058} a^{15} - \frac{8403131876578808492063898965378090648}{5063795675041016911997630357966254590609703081029} a^{14} - \frac{368131957691985979434598972179551405504545}{5063795675041016911997630357966254590609703081029} a^{13} - \frac{2041743321003946007743804129159104100949726}{5063795675041016911997630357966254590609703081029} a^{12} + \frac{38816589958416990443717062730912510541975125}{10127591350082033823995260715932509181219406162058} a^{11} + \frac{3208820685536655375310601556303428696029441}{41169070528788755382094555755823208053737423423} a^{10} + \frac{11137505358398356710112684294553822772297308}{69367064041657765917775758328304857405612370973} a^{9} - \frac{21376409818993961694473562284511182556959573}{1125287927786892647110584523992501020135489573562} a^{8} + \frac{9348368023137023396979536547102713363254729697}{1125287927786892647110584523992501020135489573562} a^{7} - \frac{3226432224324804481720612408063626769518972056}{62515995988160702617254695777361167785304976309} a^{6} - \frac{3009001595006449709968861291552906490760097592}{62515995988160702617254695777361167785304976309} a^{5} + \frac{489760704702772583261919969682341744481118641}{4630814517641533527204051539063790206318887134} a^{4} + \frac{3261558550852927134007977763702262840769244649}{6946221776462300290806077308595685309478330701} a^{3} - \frac{81640901005331092122299567014429231737795406}{257267473202307418178002863281321678128827063} a^{2} + \frac{79923749139466004885182091213017388202847817}{171511648801538278785335242187547785419218042} a + \frac{56625503755338224230229911774235636341808}{1058713881490977029539106433256467811229741}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  $11$
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:  \( 6911046034430000000 \) (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

\(\Q(\sqrt{37}) \), \(\Q(\sqrt{2701}) \), \(\Q(\sqrt{73}) \), \(\Q(\sqrt{37}, \sqrt{73})\), 8.8.388282220975269366201.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$37$37.8.7.1$x^{8} - 37$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
37.8.7.1$x^{8} - 37$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$73$73.8.7.4$x^{8} - 1140625$$8$$1$$7$$C_8$$[\ ]_{8}$
73.8.7.2$x^{8} - 1825$$8$$1$$7$$C_8$$[\ ]_{8}$