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Properties

Label 16.8.16808696443...8353.2
Degree $16$
Signature $[8, 4]$
Discriminant $61^{12}\cdot 97^{15}$
Root discriminant $1590.73$
Ramified primes $61, 97$
Class number $16$ (GRH)
Class group $[2, 2, 4]$ (GRH)
Galois group $C_2^3.C_8$ (as 16T104)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4011939306776803313, 1331571217615502528, 250647316750174834, 18985774070078141, -1423478158368283, -202418022839255, -12675582312061, -481785086219, 57917196094, 3434567043, 78893665, 2184995, -454109, -11673, -41, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 41*x^14 - 11673*x^13 - 454109*x^12 + 2184995*x^11 + 78893665*x^10 + 3434567043*x^9 + 57917196094*x^8 - 481785086219*x^7 - 12675582312061*x^6 - 202418022839255*x^5 - 1423478158368283*x^4 + 18985774070078141*x^3 + 250647316750174834*x^2 + 1331571217615502528*x + 4011939306776803313)
 
gp: K = bnfinit(x^16 - 4*x^15 - 41*x^14 - 11673*x^13 - 454109*x^12 + 2184995*x^11 + 78893665*x^10 + 3434567043*x^9 + 57917196094*x^8 - 481785086219*x^7 - 12675582312061*x^6 - 202418022839255*x^5 - 1423478158368283*x^4 + 18985774070078141*x^3 + 250647316750174834*x^2 + 1331571217615502528*x + 4011939306776803313, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 41 x^{14} - 11673 x^{13} - 454109 x^{12} + 2184995 x^{11} + 78893665 x^{10} + 3434567043 x^{9} + 57917196094 x^{8} - 481785086219 x^{7} - 12675582312061 x^{6} - 202418022839255 x^{5} - 1423478158368283 x^{4} + 18985774070078141 x^{3} + 250647316750174834 x^{2} + 1331571217615502528 x + 4011939306776803313 \)

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:  \(1680869644357963641125481571233506843563004905118353=61^{12}\cdot 97^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1590.73$
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:  $61, 97$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{61} a^{8} - \frac{2}{61} a^{7} + \frac{8}{61} a^{6} + \frac{5}{61} a^{5} - \frac{4}{61} a^{4} - \frac{30}{61} a^{3} - \frac{17}{61} a^{2} + \frac{5}{61} a + \frac{15}{61}$, $\frac{1}{61} a^{9} + \frac{4}{61} a^{7} + \frac{21}{61} a^{6} + \frac{6}{61} a^{5} + \frac{23}{61} a^{4} - \frac{16}{61} a^{3} - \frac{29}{61} a^{2} + \frac{25}{61} a + \frac{30}{61}$, $\frac{1}{122} a^{10} - \frac{1}{122} a^{9} - \frac{18}{61} a^{7} + \frac{7}{61} a^{6} + \frac{29}{61} a^{5} - \frac{23}{122} a^{4} - \frac{15}{122} a^{3} - \frac{1}{2} a^{2} - \frac{15}{122} a - \frac{29}{122}$, $\frac{1}{122} a^{11} - \frac{1}{122} a^{9} + \frac{14}{61} a^{7} - \frac{3}{61} a^{6} - \frac{29}{122} a^{5} - \frac{30}{61} a^{4} - \frac{29}{61} a^{3} + \frac{22}{61} a^{2} + \frac{7}{61} a + \frac{23}{122}$, $\frac{1}{244} a^{12} + \frac{1}{244} a^{9} - \frac{25}{61} a^{7} + \frac{47}{244} a^{6} - \frac{2}{61} a^{5} + \frac{77}{244} a^{4} - \frac{17}{244} a^{3} - \frac{117}{244} a^{2} - \frac{41}{122} a + \frac{99}{244}$, $\frac{1}{244} a^{13} - \frac{1}{244} a^{10} - \frac{1}{122} a^{9} - \frac{97}{244} a^{7} - \frac{13}{61} a^{6} - \frac{51}{244} a^{5} + \frac{25}{244} a^{4} - \frac{95}{244} a^{3} - \frac{20}{61} a^{2} + \frac{41}{244} a - \frac{13}{122}$, $\frac{1}{5861801512399528} a^{14} - \frac{6304407166503}{5861801512399528} a^{13} - \frac{741635802563}{5861801512399528} a^{12} + \frac{15408190247589}{5861801512399528} a^{11} - \frac{186613440603}{96095106760648} a^{10} + \frac{39270606570037}{5861801512399528} a^{9} + \frac{11603345581079}{5861801512399528} a^{8} + \frac{97630199397451}{5861801512399528} a^{7} - \frac{157032338067417}{1465450378099882} a^{6} - \frac{2708427411852}{732725189049941} a^{5} - \frac{1728206603910393}{5861801512399528} a^{4} + \frac{16488542188324}{732725189049941} a^{3} + \frac{51913473292184}{732725189049941} a^{2} - \frac{1762400602475111}{5861801512399528} a - \frac{977123559605033}{5861801512399528}$, $\frac{1}{44127778023069240308689389357164956359151655772246379406663262606472258015244193786964193136198841592} a^{15} - \frac{271717016890465127791125238736333378692994853747858134223376431961937592048028347028}{5515972252883655038586173669645619544893956971530797425832907825809032251905524223370524142024855199} a^{14} + \frac{11052187852905083492762633843250266451390363053551811240935300076531328973401786668420937765398285}{22063889011534620154344694678582478179575827886123189703331631303236129007622096893482096568099420796} a^{13} + \frac{5004589749476840083693791239183176048748457979988246753724662525535519487311854000256804481436979}{22063889011534620154344694678582478179575827886123189703331631303236129007622096893482096568099420796} a^{12} - \frac{28056645903520839458403063065619989404639543729241156659085615149752249083832252453819896506009173}{11031944505767310077172347339291239089787913943061594851665815651618064503811048446741048284049710398} a^{11} - \frac{22899701906242305108454514397169363129001152296483770686945562268743978694285609730642352346761335}{22063889011534620154344694678582478179575827886123189703331631303236129007622096893482096568099420796} a^{10} - \frac{3155292461198937268363600725417085647374551882150293646217429725286233239254163546736730771396826}{5515972252883655038586173669645619544893956971530797425832907825809032251905524223370524142024855199} a^{9} - \frac{63581667467043305301397256607785032485143300450996390273416828214230902104242874060779057365098141}{11031944505767310077172347339291239089787913943061594851665815651618064503811048446741048284049710398} a^{8} - \frac{8685521548088973493975182658485240159473594164681187451253124423243444710473492134758529447934858637}{44127778023069240308689389357164956359151655772246379406663262606472258015244193786964193136198841592} a^{7} + \frac{8967470219097910701251636953753086614013956022087452768001170297725733942236456064382131245827470479}{22063889011534620154344694678582478179575827886123189703331631303236129007622096893482096568099420796} a^{6} + \frac{17928603774923401193519337958098121571827024692877266699055010958714573713663585605564001676077042837}{44127778023069240308689389357164956359151655772246379406663262606472258015244193786964193136198841592} a^{5} - \frac{8514802765860578137223475867198019627027639754433499800621381624101733983855528837867424892150110879}{44127778023069240308689389357164956359151655772246379406663262606472258015244193786964193136198841592} a^{4} - \frac{5212941345533274786104044699402879923959463827477961127612273225288391740254976645290580894226791021}{11031944505767310077172347339291239089787913943061594851665815651618064503811048446741048284049710398} a^{3} - \frac{18124903683201598754916864831870906899959145265357161901280148883502592743031990184605056307034790193}{44127778023069240308689389357164956359151655772246379406663262606472258015244193786964193136198841592} a^{2} + \frac{47976917346209916744013909197472666879795488241121603111294179268579498935434653297244733712846089}{11031944505767310077172347339291239089787913943061594851665815651618064503811048446741048284049710398} a + \frac{2806110871176835737290681611957438758248811533885408122094806746286028158567516753900709805927696587}{44127778023069240308689389357164956359151655772246379406663262606472258015244193786964193136198841592}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}$, 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:  $11$
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:  \( 2572468789910000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_8$ (as 16T104):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 22 conjugacy class representatives for $C_2^3.C_8$
Character table for $C_2^3.C_8$ is not computed

Intermediate fields

\(\Q(\sqrt{97}) \), 4.4.912673.1, 8.8.1118720199956720578033.1

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

Sibling fields

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ $16$ $16$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ $16$

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
61Data not computed
97Data not computed