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Properties

Label 16.0.68989390528...3049.5
Degree $16$
Signature $[0, 8]$
Discriminant $11^{12}\cdot 89^{13}$
Root discriminant $231.70$
Ramified primes $11, 89$
Class number $16$ (GRH)
Class group $[4, 4]$ (GRH)
Galois group $C_4.D_4:C_4$ (as 16T260)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![288557906703, -211399377596, 44950407942, -11902179377, 5518502641, -992861822, 251387127, -65792209, 8165048, -1530825, 269865, -16062, 2537, -413, -24, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 24*x^14 - 413*x^13 + 2537*x^12 - 16062*x^11 + 269865*x^10 - 1530825*x^9 + 8165048*x^8 - 65792209*x^7 + 251387127*x^6 - 992861822*x^5 + 5518502641*x^4 - 11902179377*x^3 + 44950407942*x^2 - 211399377596*x + 288557906703)
 
gp: K = bnfinit(x^16 - 24*x^14 - 413*x^13 + 2537*x^12 - 16062*x^11 + 269865*x^10 - 1530825*x^9 + 8165048*x^8 - 65792209*x^7 + 251387127*x^6 - 992861822*x^5 + 5518502641*x^4 - 11902179377*x^3 + 44950407942*x^2 - 211399377596*x + 288557906703, 1)
 

Normalized defining polynomial

\( x^{16} - 24 x^{14} - 413 x^{13} + 2537 x^{12} - 16062 x^{11} + 269865 x^{10} - 1530825 x^{9} + 8165048 x^{8} - 65792209 x^{7} + 251387127 x^{6} - 992861822 x^{5} + 5518502641 x^{4} - 11902179377 x^{3} + 44950407942 x^{2} - 211399377596 x + 288557906703 \)

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:  \(68989390528102141148007963042095383049=11^{12}\cdot 89^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $231.70$
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:  $11, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{6} a^{13} + \frac{1}{6} a^{12} + \frac{1}{3} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{3} a^{8} + \frac{1}{6} a^{7} - \frac{1}{6} a^{6} - \frac{1}{3} a^{5} + \frac{1}{6} a^{4} - \frac{1}{2} a^{3} - \frac{1}{3} a^{2} + \frac{1}{6} a - \frac{1}{2}$, $\frac{1}{96} a^{14} + \frac{1}{24} a^{13} + \frac{41}{96} a^{12} - \frac{7}{32} a^{11} - \frac{3}{16} a^{10} - \frac{11}{96} a^{9} + \frac{43}{96} a^{8} + \frac{1}{12} a^{7} + \frac{19}{96} a^{6} - \frac{29}{96} a^{5} + \frac{1}{16} a^{4} + \frac{13}{96} a^{3} + \frac{43}{96} a^{2} + \frac{1}{4} a + \frac{11}{32}$, $\frac{1}{383068515740269295028210755899793755926760223406528228714717174236048783841216} a^{15} + \frac{535619474042012332577552502419368994694425964047889188472706189611419903551}{383068515740269295028210755899793755926760223406528228714717174236048783841216} a^{14} - \frac{21889796488384175846127300439093028240897825841920649092340667148007940054475}{383068515740269295028210755899793755926760223406528228714717174236048783841216} a^{13} + \frac{21236608611011149792440765557893559501988700506540499772803789047443581595599}{191534257870134647514105377949896877963380111703264114357358587118024391920608} a^{12} - \frac{103440293324483079667863061371102305105709026536861593320766028956079272845129}{383068515740269295028210755899793755926760223406528228714717174236048783841216} a^{11} + \frac{73060042287686698762676342383642862689687528967686191183269914029060825537871}{383068515740269295028210755899793755926760223406528228714717174236048783841216} a^{10} + \frac{25909325183402433822507848497056599580281595444310774760088975012990554555489}{191534257870134647514105377949896877963380111703264114357358587118024391920608} a^{9} + \frac{65750504124932888681402854240880809073970193979724015024267671255065439004529}{383068515740269295028210755899793755926760223406528228714717174236048783841216} a^{8} - \frac{6659708529606677840460250912706898099524023796763150151652096054908647760487}{127689505246756431676070251966597918642253407802176076238239058078682927947072} a^{7} + \frac{42515899061430830162521576651421733018979367404970909445118616607958315336193}{95767128935067323757052688974948438981690055851632057178679293559012195960304} a^{6} - \frac{18399414899734742460855501875479482505378761041326217066375965243849678467241}{383068515740269295028210755899793755926760223406528228714717174236048783841216} a^{5} + \frac{46011385445352723312907644416655563474741645976935728245720841433024488243589}{127689505246756431676070251966597918642253407802176076238239058078682927947072} a^{4} + \frac{43570222947818964495263305449988286582417074209020553257660909071207820154197}{191534257870134647514105377949896877963380111703264114357358587118024391920608} a^{3} + \frac{81313008789989873581015925168626611557596833237022749952784095442196598489985}{383068515740269295028210755899793755926760223406528228714717174236048783841216} a^{2} + \frac{9475502198201497496496659548914548258783045607416025675482670100819433480201}{383068515740269295028210755899793755926760223406528228714717174236048783841216} a + \frac{6563498763358588842364996096496493262232967663620181878352157152756770910025}{127689505246756431676070251966597918642253407802176076238239058078682927947072}$

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

Class group and class number

$C_{4}\times C_{4}$, which has order $16$ (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:  \( 946035934551 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4.D_4:C_4$ (as 16T260):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 32 conjugacy class representatives for $C_4.D_4:C_4$
Character table for $C_4.D_4:C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-11}) \), 4.0.10769.1, 8.0.81756214392809.2

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

Sibling fields

Degree 16 sibling: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ $16$ R $16$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}$ $16$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$ $16$ $16$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
$11$11.4.3.1$x^{4} + 33$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
11.4.3.1$x^{4} + 33$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
11.4.3.1$x^{4} + 33$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
11.4.3.1$x^{4} + 33$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
$89$89.4.3.4$x^{4} + 2403$$4$$1$$3$$C_4$$[\ ]_{4}$
89.4.3.3$x^{4} + 267$$4$$1$$3$$C_4$$[\ ]_{4}$
89.8.7.4$x^{8} - 64881$$8$$1$$7$$C_8$$[\ ]_{8}$