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Properties

Label 16.0.16459410512...7801.6
Degree $16$
Signature $[0, 8]$
Discriminant $41^{13}\cdot 59^{12}$
Root discriminant $435.04$
Ramified primes $41, 59$
Class number $240$ (GRH)
Class group $[4, 60]$ (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![281742122142543, 23625665057905, -28777843094468, -4239496133880, 1496275591245, 411196014948, 11182903017, -5933432448, -428796243, 59571195, 9936127, 290065, -26089, -219, 40, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 + 40*x^14 - 219*x^13 - 26089*x^12 + 290065*x^11 + 9936127*x^10 + 59571195*x^9 - 428796243*x^8 - 5933432448*x^7 + 11182903017*x^6 + 411196014948*x^5 + 1496275591245*x^4 - 4239496133880*x^3 - 28777843094468*x^2 + 23625665057905*x + 281742122142543)
 
gp: K = bnfinit(x^16 - 2*x^15 + 40*x^14 - 219*x^13 - 26089*x^12 + 290065*x^11 + 9936127*x^10 + 59571195*x^9 - 428796243*x^8 - 5933432448*x^7 + 11182903017*x^6 + 411196014948*x^5 + 1496275591245*x^4 - 4239496133880*x^3 - 28777843094468*x^2 + 23625665057905*x + 281742122142543, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} + 40 x^{14} - 219 x^{13} - 26089 x^{12} + 290065 x^{11} + 9936127 x^{10} + 59571195 x^{9} - 428796243 x^{8} - 5933432448 x^{7} + 11182903017 x^{6} + 411196014948 x^{5} + 1496275591245 x^{4} - 4239496133880 x^{3} - 28777843094468 x^{2} + 23625665057905 x + 281742122142543 \)

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:  \(1645941051244254531415289228525803373267801=41^{13}\cdot 59^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $435.04$
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:  $41, 59$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{41} a^{12} - \frac{18}{41} a^{11} - \frac{14}{41} a^{10} + \frac{1}{41} a^{9} + \frac{9}{41} a^{8} - \frac{12}{41} a^{7} + \frac{14}{41} a^{6} + \frac{7}{41} a^{5} + \frac{11}{41} a^{4} + \frac{10}{41} a^{3} + \frac{14}{41} a^{2} + \frac{1}{41} a + \frac{18}{41}$, $\frac{1}{123} a^{13} - \frac{1}{123} a^{12} + \frac{49}{123} a^{11} + \frac{50}{123} a^{10} - \frac{5}{41} a^{9} + \frac{59}{123} a^{8} - \frac{26}{123} a^{7} - \frac{14}{41} a^{6} - \frac{34}{123} a^{5} - \frac{8}{123} a^{4} - \frac{7}{41} a^{3} - \frac{7}{123} a^{2} - \frac{47}{123} a + \frac{20}{41}$, $\frac{1}{123} a^{14} - \frac{7}{41} a^{11} - \frac{31}{123} a^{10} - \frac{4}{123} a^{9} - \frac{10}{41} a^{8} + \frac{16}{123} a^{7} - \frac{10}{123} a^{6} - \frac{3}{41} a^{5} + \frac{58}{123} a^{4} - \frac{16}{123} a^{3} + \frac{4}{41} a^{2} - \frac{35}{123} a + \frac{19}{41}$, $\frac{1}{2013903376332343088152260627825966566698466822820966388395651474115760572994069720741525755005431973} a^{15} - \frac{2486999171036915533939154314624564561626128365624200055965736670624055709760577261711210482072413}{2013903376332343088152260627825966566698466822820966388395651474115760572994069720741525755005431973} a^{14} - \frac{863822609834494977739250378853700974352761214742208753498003440586346299889111824338390607620789}{671301125444114362717420209275322188899488940940322129465217158038586857664689906913841918335143991} a^{13} - \frac{780314130359378394393893365466018558844880843023996213071653532520619676751868896812027124062904}{74589013938234929190824467697258020988832104548924681051690795337620761962743322990426879815015999} a^{12} - \frac{409010935309520391226496023725118992010940121494399274429608326547743887297566221036043450425680073}{2013903376332343088152260627825966566698466822820966388395651474115760572994069720741525755005431973} a^{11} - \frac{245820619515638894531176659703104902140577119060979288246681253095173645486026940834188375693904658}{671301125444114362717420209275322188899488940940322129465217158038586857664689906913841918335143991} a^{10} - \frac{94255253066492107159561315743818743782046452345668884157298516689643439677388234313596367872769492}{2013903376332343088152260627825966566698466822820966388395651474115760572994069720741525755005431973} a^{9} + \frac{404059820398381260006465518231350952387170192951551335263682362296352054368715236034167157379302087}{2013903376332343088152260627825966566698466822820966388395651474115760572994069720741525755005431973} a^{8} + \frac{687888989094683276782688112943807404582270828167746369364853363776745222859126402216676598728861758}{2013903376332343088152260627825966566698466822820966388395651474115760572994069720741525755005431973} a^{7} - \frac{738070732210890065118739446760023226868596791416036297450743417311718942146849323306509241020092995}{2013903376332343088152260627825966566698466822820966388395651474115760572994069720741525755005431973} a^{6} + \frac{973672021476906287706631212640904035678262061172936699990598747365606267916741619534861616273017589}{2013903376332343088152260627825966566698466822820966388395651474115760572994069720741525755005431973} a^{5} + \frac{369045369104130752103478540280955604103688101404484786116856458592847144468991394078361695333733370}{2013903376332343088152260627825966566698466822820966388395651474115760572994069720741525755005431973} a^{4} - \frac{493330863427732415122701619362027908004354240663002994970435503630492757521372691429808979111312629}{2013903376332343088152260627825966566698466822820966388395651474115760572994069720741525755005431973} a^{3} + \frac{948624324937869510929585609768347085089829620287452704682900016011340878754997415655867199355337370}{2013903376332343088152260627825966566698466822820966388395651474115760572994069720741525755005431973} a^{2} + \frac{714597572559151300254090980819774888235944668573434746033349508644268557650276208356053663872649475}{2013903376332343088152260627825966566698466822820966388395651474115760572994069720741525755005431973} a - \frac{252991678805341010034331290420855959094769203227403365561751937400504165182850874317024308677622573}{671301125444114362717420209275322188899488940940322129465217158038586857664689906913841918335143991}$

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

Class group and class number

$C_{4}\times C_{60}$, which has order $240$ (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:  \( 5435428829400 \) (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

\(\Q(\sqrt{-59}) \), 4.0.142721.1, 8.0.1403871411605561.2

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}$ R

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
$41$41.4.3.4$x^{4} + 8856$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.3.3$x^{4} + 246$$4$$1$$3$$C_4$$[\ ]_{4}$
41.8.7.2$x^{8} - 1476$$8$$1$$7$$C_8$$[\ ]_{8}$
$59$59.8.6.2$x^{8} + 177 x^{4} + 13924$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
59.8.6.2$x^{8} + 177 x^{4} + 13924$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$