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Properties

Label 16.0.29786305681...8976.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{66}\cdot 7^{6}\cdot 193^{4}\cdot 223^{4}$
Root discriminant $521.35$
Ramified primes $2, 7, 193, 223$
Class number $81864483840$ (GRH)
Class group $[2, 2, 2, 2, 2, 2, 2, 6, 106594380]$ (GRH)
Galois group $(C_2^2\times C_4).C_2^4$ (as 16T471)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![55151228745540959074, -5494406389961891840, 2467664760780016224, -109475242669266480, 43925474650991112, -892387439161696, 358605823359368, -3619337419376, 1528170097824, -7631842416, 3643957168, -7958096, 4898708, -3216, 3464, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 3464*x^14 - 3216*x^13 + 4898708*x^12 - 7958096*x^11 + 3643957168*x^10 - 7631842416*x^9 + 1528170097824*x^8 - 3619337419376*x^7 + 358605823359368*x^6 - 892387439161696*x^5 + 43925474650991112*x^4 - 109475242669266480*x^3 + 2467664760780016224*x^2 - 5494406389961891840*x + 55151228745540959074)
 
gp: K = bnfinit(x^16 + 3464*x^14 - 3216*x^13 + 4898708*x^12 - 7958096*x^11 + 3643957168*x^10 - 7631842416*x^9 + 1528170097824*x^8 - 3619337419376*x^7 + 358605823359368*x^6 - 892387439161696*x^5 + 43925474650991112*x^4 - 109475242669266480*x^3 + 2467664760780016224*x^2 - 5494406389961891840*x + 55151228745540959074, 1)
 

Normalized defining polynomial

\( x^{16} + 3464 x^{14} - 3216 x^{13} + 4898708 x^{12} - 7958096 x^{11} + 3643957168 x^{10} - 7631842416 x^{9} + 1528170097824 x^{8} - 3619337419376 x^{7} + 358605823359368 x^{6} - 892387439161696 x^{5} + 43925474650991112 x^{4} - 109475242669266480 x^{3} + 2467664760780016224 x^{2} - 5494406389961891840 x + 55151228745540959074 \)

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:  \(29786305681426893421813381491703267748478976=2^{66}\cdot 7^{6}\cdot 193^{4}\cdot 223^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $521.35$
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:  $2, 7, 193, 223$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{31} a^{13} - \frac{14}{31} a^{12} + \frac{4}{31} a^{11} - \frac{12}{31} a^{10} - \frac{12}{31} a^{9} + \frac{4}{31} a^{8} + \frac{2}{31} a^{7} - \frac{14}{31} a^{6} - \frac{2}{31} a^{5} + \frac{8}{31} a^{4} + \frac{13}{31} a^{3} + \frac{11}{31} a^{2} + \frac{2}{31} a - \frac{11}{31}$, $\frac{1}{31} a^{14} - \frac{6}{31} a^{12} + \frac{13}{31} a^{11} + \frac{6}{31} a^{10} - \frac{9}{31} a^{9} - \frac{4}{31} a^{8} + \frac{14}{31} a^{7} - \frac{12}{31} a^{6} + \frac{11}{31} a^{5} + \frac{1}{31} a^{4} + \frac{7}{31} a^{3} + \frac{1}{31} a^{2} - \frac{14}{31} a + \frac{1}{31}$, $\frac{1}{334047461538252809768690096930513245764057866534690335581153088148227457323725437947754356663889856812200864676395828176804353} a^{15} + \frac{1918758438196215184329787524271720422367151407168402139731251069518108193586880161383601210346208463573777871827941809601015}{334047461538252809768690096930513245764057866534690335581153088148227457323725437947754356663889856812200864676395828176804353} a^{14} + \frac{699733276186947708607685790528861782858062631329336731857700706143601909963756551022088807128122310420633849337972335344859}{47721065934036115681241442418644749394865409504955762225879012592603922474817919706822050951984265258885837810913689739543479} a^{13} + \frac{44933447004961452194177875973425026804935424675686056271714331876402132154105015690125396413072533282533090038795358605544636}{334047461538252809768690096930513245764057866534690335581153088148227457323725437947754356663889856812200864676395828176804353} a^{12} + \frac{12189777019096700112821539809552988309978423207071782252212976763481849948207805508009624737975086095988592433485986261846629}{47721065934036115681241442418644749394865409504955762225879012592603922474817919706822050951984265258885837810913689739543479} a^{11} - \frac{39631444240266043082971517364907040271448793418081750263101506350194907411980363532652132403026163832923643992914928148416332}{334047461538252809768690096930513245764057866534690335581153088148227457323725437947754356663889856812200864676395828176804353} a^{10} + \frac{4169011171916221786264816477720519586289740449194599571523929981055500290816929578494962907985165358034332028928982462667156}{334047461538252809768690096930513245764057866534690335581153088148227457323725437947754356663889856812200864676395828176804353} a^{9} - \frac{69590844845671528520483857157185998537009511078783869965995485582157301327717359986449943113833679003592073843189131044403568}{334047461538252809768690096930513245764057866534690335581153088148227457323725437947754356663889856812200864676395828176804353} a^{8} - \frac{8289648306620256950768303023234903696098013459190655918937475647751023993186729094894308772959952370380834155552297723014229}{19649850678720753515805299819441955633179874502040607975361946361660438666101496349867903333169991577188286157435048716282609} a^{7} - \frac{53521362663231611488383524958376193011369337053873803027840464169069902386569495404027343346162248971526198019119846320461009}{334047461538252809768690096930513245764057866534690335581153088148227457323725437947754356663889856812200864676395828176804353} a^{6} + \frac{21396664222042567162317280017312869792944893965473918153744219338025372945867890925717045603868786529296879258315242544912596}{47721065934036115681241442418644749394865409504955762225879012592603922474817919706822050951984265258885837810913689739543479} a^{5} - \frac{25529412232691054837980726661238138760366129673849481999939710840067874784321695372882207082704060817657765685925631733378808}{334047461538252809768690096930513245764057866534690335581153088148227457323725437947754356663889856812200864676395828176804353} a^{4} - \frac{92972894256360270243212519507394386986137669280609768835989777666336569759729328268782872592055686525192430232833719461212448}{334047461538252809768690096930513245764057866534690335581153088148227457323725437947754356663889856812200864676395828176804353} a^{3} - \frac{99270196819521977143776493060835093270731954135442459354311279584688586708584941993690221799858408603838995609283813155283237}{334047461538252809768690096930513245764057866534690335581153088148227457323725437947754356663889856812200864676395828176804353} a^{2} - \frac{25308523584188174745943512334174184367546355882957136890092071854162867977058240528927357747505160069208966194692801530517989}{334047461538252809768690096930513245764057866534690335581153088148227457323725437947754356663889856812200864676395828176804353} a + \frac{82194172805141790760940330307538983891467658382003017081640006174444096539226723100902732290061297715162807150008380683290510}{334047461538252809768690096930513245764057866534690335581153088148227457323725437947754356663889856812200864676395828176804353}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{6}\times C_{106594380}$, which has order $81864483840$ (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:  \( 20726.065235 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2^2\times C_4).C_2^4$ (as 16T471):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 46 conjugacy class representatives for $(C_2^2\times C_4).C_2^4$
Character table for $(C_2^2\times C_4).C_2^4$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.14336.1, \(\Q(\zeta_{16})^+\), 4.4.7168.1, 8.8.3288334336.1

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 R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
2Data not computed
$7$7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.8.6.2$x^{8} - 49 x^{4} + 3969$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
193Data not computed
223Data not computed