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Properties

Label 16.8.12054181652...3369.3
Degree $16$
Signature $[8, 4]$
Discriminant $61^{14}\cdot 73^{14}$
Root discriminant $1558.01$
Ramified primes $61, 73$
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![485904167550453131, 57598242321428699, 20824625479139251, 2670926156166982, -146350495119186, -33456378639173, -2434234654366, -62119510308, 15244519653, 1097370602, 34740094, 418582, -243786, -8906, -67, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 67*x^14 - 8906*x^13 - 243786*x^12 + 418582*x^11 + 34740094*x^10 + 1097370602*x^9 + 15244519653*x^8 - 62119510308*x^7 - 2434234654366*x^6 - 33456378639173*x^5 - 146350495119186*x^4 + 2670926156166982*x^3 + 20824625479139251*x^2 + 57598242321428699*x + 485904167550453131)
 
gp: K = bnfinit(x^16 - 67*x^14 - 8906*x^13 - 243786*x^12 + 418582*x^11 + 34740094*x^10 + 1097370602*x^9 + 15244519653*x^8 - 62119510308*x^7 - 2434234654366*x^6 - 33456378639173*x^5 - 146350495119186*x^4 + 2670926156166982*x^3 + 20824625479139251*x^2 + 57598242321428699*x + 485904167550453131, 1)
 

Normalized defining polynomial

\( x^{16} - 67 x^{14} - 8906 x^{13} - 243786 x^{12} + 418582 x^{11} + 34740094 x^{10} + 1097370602 x^{9} + 15244519653 x^{8} - 62119510308 x^{7} - 2434234654366 x^{6} - 33456378639173 x^{5} - 146350495119186 x^{4} + 2670926156166982 x^{3} + 20824625479139251 x^{2} + 57598242321428699 x + 485904167550453131 \)

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:  \(1205418165202307934078936492069912344967184840483369=61^{14}\cdot 73^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1558.01$
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:  $61, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{156521} a^{14} + \frac{61262}{156521} a^{13} - \frac{47099}{156521} a^{12} - \frac{13075}{156521} a^{11} - \frac{37214}{156521} a^{10} - \frac{17892}{156521} a^{9} + \frac{40273}{156521} a^{8} + \frac{71543}{156521} a^{7} - \frac{68709}{156521} a^{6} + \frac{42357}{156521} a^{5} + \frac{61639}{156521} a^{4} + \frac{33608}{156521} a^{3} - \frac{15222}{156521} a^{2} - \frac{63683}{156521} a - \frac{72651}{156521}$, $\frac{1}{504678532167567305174560851333574116030874590402737433168808679700581179135382013772350464179739170631842153245576313} a^{15} + \frac{646005418344157616088655921664303307809601409008259253240309506917961806181414632938516930685246793002245518837}{504678532167567305174560851333574116030874590402737433168808679700581179135382013772350464179739170631842153245576313} a^{14} + \frac{126442892527682179406187796837770108254717592612976972075172292576701228237485250842071811495880781078695604472710957}{504678532167567305174560851333574116030874590402737433168808679700581179135382013772350464179739170631842153245576313} a^{13} - \frac{112618997484471012137162311207742152861201253971136180462200448317105215602953818302811267986203094453435757982170574}{504678532167567305174560851333574116030874590402737433168808679700581179135382013772350464179739170631842153245576313} a^{12} + \frac{188782683887237318937417319416475726241017726024845079961832012881359109867898086951898325467340495700907151173625564}{504678532167567305174560851333574116030874590402737433168808679700581179135382013772350464179739170631842153245576313} a^{11} + \frac{215603667861821198665407979279674870861533897242782385949606844736491883825004814940637759721637872963535517673191375}{504678532167567305174560851333574116030874590402737433168808679700581179135382013772350464179739170631842153245576313} a^{10} - \frac{44226863122963833031167871062509805591328751551591297896800624417469303991025300608983767253376946571795642593604209}{504678532167567305174560851333574116030874590402737433168808679700581179135382013772350464179739170631842153245576313} a^{9} + \frac{91199357644416763275878797043077477091998701802448929469585137593933194307454324161904380619155078804334118639552373}{504678532167567305174560851333574116030874590402737433168808679700581179135382013772350464179739170631842153245576313} a^{8} + \frac{125864031841518853093347197464353794971735163537679069691779702778426737041534116042851819626045795849816197607126585}{504678532167567305174560851333574116030874590402737433168808679700581179135382013772350464179739170631842153245576313} a^{7} + \frac{159120391468953180148850671885921056785084267457692371405571682040240261884137918990782083849631004945065030512883736}{504678532167567305174560851333574116030874590402737433168808679700581179135382013772350464179739170631842153245576313} a^{6} - \frac{250399563362148226019456426021654415786649312691182244586155507118983334627557152582040835465522726083722114883485962}{504678532167567305174560851333574116030874590402737433168808679700581179135382013772350464179739170631842153245576313} a^{5} + \frac{51405787247852686755645018558962943791540382286687199503197668974991986812862288982504981758640364368219473601902946}{504678532167567305174560851333574116030874590402737433168808679700581179135382013772350464179739170631842153245576313} a^{4} + \frac{111263702319709761846202276130650163359560961337047762240854388944215982417232774482673437749634510360452396834017059}{504678532167567305174560851333574116030874590402737433168808679700581179135382013772350464179739170631842153245576313} a^{3} + \frac{169103252758168212026693883254341059902584952912954419106815023986808057471756514779062594103377955098194667530429136}{504678532167567305174560851333574116030874590402737433168808679700581179135382013772350464179739170631842153245576313} a^{2} - \frac{195539960142878024972110624334786695449111873200537335408174486145699577135067527050827303939207641182403119028221047}{504678532167567305174560851333574116030874590402737433168808679700581179135382013772350464179739170631842153245576313} a + \frac{92446817361294930147924667534017613938978401944633250994007468489482870170180371968144818775048759683091039924313380}{504678532167567305174560851333574116030874590402737433168808679700581179135382013772350464179739170631842153245576313}$

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:  \( 8246406792500000000 \) (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

\(\Q(\sqrt{61}) \), \(\Q(\sqrt{4453}) \), \(\Q(\sqrt{73}) \), \(\Q(\sqrt{61}, \sqrt{73})\), 8.8.7796795992041567776329.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$61$61.8.7.1$x^{8} - 61$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
61.8.7.1$x^{8} - 61$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$73$73.8.7.2$x^{8} - 1825$$8$$1$$7$$C_8$$[\ ]_{8}$
73.8.7.2$x^{8} - 1825$$8$$1$$7$$C_8$$[\ ]_{8}$