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Properties

Label 16.0.81925410828...5625.5
Degree $16$
Signature $[0, 8]$
Discriminant $5^{14}\cdot 41^{10}$
Root discriminant $41.65$
Ramified primes $5, 41$
Class number $8$ (GRH)
Class group $[2, 2, 2]$ (GRH)
Galois group $(C_2\times C_4).D_4$ (as 16T121)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![173852905, 58427950, 100016220, 24663705, 25816210, 4439000, 3953115, 430260, 408181, 21851, 29552, 13, 1450, -53, 52, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 52*x^14 - 53*x^13 + 1450*x^12 + 13*x^11 + 29552*x^10 + 21851*x^9 + 408181*x^8 + 430260*x^7 + 3953115*x^6 + 4439000*x^5 + 25816210*x^4 + 24663705*x^3 + 100016220*x^2 + 58427950*x + 173852905)
 
gp: K = bnfinit(x^16 - x^15 + 52*x^14 - 53*x^13 + 1450*x^12 + 13*x^11 + 29552*x^10 + 21851*x^9 + 408181*x^8 + 430260*x^7 + 3953115*x^6 + 4439000*x^5 + 25816210*x^4 + 24663705*x^3 + 100016220*x^2 + 58427950*x + 173852905, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 52 x^{14} - 53 x^{13} + 1450 x^{12} + 13 x^{11} + 29552 x^{10} + 21851 x^{9} + 408181 x^{8} + 430260 x^{7} + 3953115 x^{6} + 4439000 x^{5} + 25816210 x^{4} + 24663705 x^{3} + 100016220 x^{2} + 58427950 x + 173852905 \)

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:  \(81925410828566900634765625=5^{14}\cdot 41^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.65$
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:  $5, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{21} a^{14} + \frac{10}{21} a^{13} + \frac{1}{3} a^{12} + \frac{1}{3} a^{11} + \frac{1}{21} a^{10} + \frac{10}{21} a^{9} + \frac{2}{21} a^{8} - \frac{5}{21} a^{7} - \frac{4}{21} a^{6} + \frac{8}{21} a^{5} + \frac{2}{7} a^{4} + \frac{1}{21} a^{3} - \frac{3}{7} a^{2} + \frac{1}{21} a + \frac{2}{21}$, $\frac{1}{353754261804557764683811036001942120350440145978316511} a^{15} + \frac{7069422758651308042155047330312977188587347797223651}{353754261804557764683811036001942120350440145978316511} a^{14} - \frac{75457347905231258174237468055837656229404923651533815}{353754261804557764683811036001942120350440145978316511} a^{13} - \frac{382714217877901007452435092441157372752936758097748}{4594211192266983956932610857168079485070651246471643} a^{12} + \frac{60730010214133343181738999600713680022901175481615749}{353754261804557764683811036001942120350440145978316511} a^{11} + \frac{156345700086142679164328996823093056051957657472209422}{353754261804557764683811036001942120350440145978316511} a^{10} - \frac{88333060913326893304573272415168077153018898993562047}{353754261804557764683811036001942120350440145978316511} a^{9} - \frac{25343564387008541103438951733047086455504072525116491}{353754261804557764683811036001942120350440145978316511} a^{8} - \frac{2968039353080845748443869601567690467342040897642063}{353754261804557764683811036001942120350440145978316511} a^{7} + \frac{9611717823659999822790169429342396770629237436713273}{353754261804557764683811036001942120350440145978316511} a^{6} + \frac{46765123818038012813657727391298407502586996683491215}{117918087268185921561270345333980706783480048659438837} a^{5} - \frac{5132218603892347803451300389716484123356702988511804}{32159478345868887698528276000176556395494558725301501} a^{4} - \frac{49833543187446107567386554519768511303900879809460221}{117918087268185921561270345333980706783480048659438837} a^{3} - \frac{17293031502080157184604440569312610937509834834539932}{353754261804557764683811036001942120350440145978316511} a^{2} + \frac{23158000838371614221431729058234864656357613388966782}{353754261804557764683811036001942120350440145978316511} a - \frac{45967819517141947639521629838205337513375539583108252}{117918087268185921561270345333980706783480048659438837}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}$, which has order $8$ (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:  \( \frac{2390187378161257230721652269441593004136}{20362077513829080949136576924869019188843324521} a^{15} - \frac{10915677070151978890860629835774065813089}{20362077513829080949136576924869019188843324521} a^{14} + \frac{143979842107559617818227089458096797849909}{20362077513829080949136576924869019188843324521} a^{13} - \frac{73492254296925227691554769220123708316002}{2908868216261297278448082417838431312691903503} a^{12} + \frac{4103429551636212105740593819808398555987067}{20362077513829080949136576924869019188843324521} a^{11} - \frac{8916877206592538401753021651368821419088539}{20362077513829080949136576924869019188843324521} a^{10} + \frac{68342324595695569973258528427973540701846964}{20362077513829080949136576924869019188843324521} a^{9} - \frac{12215095383885761729807235968705801691824065}{2908868216261297278448082417838431312691903503} a^{8} + \frac{770923906869181389665772194692262733482979514}{20362077513829080949136576924869019188843324521} a^{7} - \frac{436400561655368565008698746120704710390740527}{20362077513829080949136576924869019188843324521} a^{6} + \frac{1970300330972500421156189799634401229620552906}{6787359171276360316378858974956339729614441507} a^{5} - \frac{517017974569416029815168543806793192542119101}{20362077513829080949136576924869019188843324521} a^{4} + \frac{9824031772702074569749651890685654491659523401}{6787359171276360316378858974956339729614441507} a^{3} + \frac{4637256044562908247553772856214123900164572972}{20362077513829080949136576924869019188843324521} a^{2} + \frac{77020486465002392904827586449600817451620482470}{20362077513829080949136576924869019188843324521} a + \frac{7652173195621244863259225182130133332003431958}{6787359171276360316378858974956339729614441507} \) (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:  \( 1132490.59957 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times C_4).D_4$ (as 16T121):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 28 conjugacy class representatives for $(C_2\times C_4).D_4$
Character table for $(C_2\times C_4).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 4.4.5125.1, 4.0.1025.1, 8.4.9051265703125.2, 8.4.9051265703125.1, 8.0.26265625.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$5$5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$41$41.4.2.2$x^{4} - 41 x^{2} + 20172$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
41.4.2.2$x^{4} - 41 x^{2} + 20172$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
41.4.3.3$x^{4} + 246$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.3.3$x^{4} + 246$$4$$1$$3$$C_4$$[\ ]_{4}$