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Properties

Label 16.0.27510376712...625.40
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 101^{12}$
Root discriminant $106.53$
Ramified primes $5, 101$
Class number $2560$ (GRH)
Class group $[2, 2, 4, 4, 40]$ (GRH)
Galois group $C_4:C_4$ (as 16T8)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![906892741, 368350701, -3939082, 64628944, 9489133, -7936374, 3496685, -48180, -156054, 76105, 4075, -3976, 563, 31, -37, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 37*x^14 + 31*x^13 + 563*x^12 - 3976*x^11 + 4075*x^10 + 76105*x^9 - 156054*x^8 - 48180*x^7 + 3496685*x^6 - 7936374*x^5 + 9489133*x^4 + 64628944*x^3 - 3939082*x^2 + 368350701*x + 906892741)
 
gp: K = bnfinit(x^16 - x^15 - 37*x^14 + 31*x^13 + 563*x^12 - 3976*x^11 + 4075*x^10 + 76105*x^9 - 156054*x^8 - 48180*x^7 + 3496685*x^6 - 7936374*x^5 + 9489133*x^4 + 64628944*x^3 - 3939082*x^2 + 368350701*x + 906892741, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 37 x^{14} + 31 x^{13} + 563 x^{12} - 3976 x^{11} + 4075 x^{10} + 76105 x^{9} - 156054 x^{8} - 48180 x^{7} + 3496685 x^{6} - 7936374 x^{5} + 9489133 x^{4} + 64628944 x^{3} - 3939082 x^{2} + 368350701 x + 906892741 \)

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:  \(275103767122062920083301025390625=5^{12}\cdot 101^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $106.53$
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:  $5, 101$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} + \frac{2}{5} a^{7} + \frac{2}{5} a^{6} - \frac{1}{5} a^{5} - \frac{1}{5} a^{4} + \frac{1}{5} a^{3} + \frac{2}{5} a^{2} - \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{9} - \frac{2}{5} a^{7} + \frac{1}{5} a^{5} - \frac{2}{5} a^{4} - \frac{1}{5} a^{2} - \frac{2}{5}$, $\frac{1}{10} a^{10} - \frac{1}{10} a^{9} - \frac{1}{10} a^{8} + \frac{2}{5} a^{7} + \frac{3}{10} a^{6} + \frac{1}{10} a^{5} - \frac{2}{5} a^{4} + \frac{3}{10} a^{2} - \frac{2}{5} a + \frac{3}{10}$, $\frac{1}{190} a^{11} - \frac{1}{190} a^{10} - \frac{9}{190} a^{9} + \frac{4}{95} a^{8} - \frac{63}{190} a^{7} + \frac{69}{190} a^{6} - \frac{43}{95} a^{5} + \frac{36}{95} a^{4} + \frac{3}{10} a^{3} + \frac{41}{95} a^{2} + \frac{7}{38} a + \frac{4}{19}$, $\frac{1}{190} a^{12} + \frac{9}{190} a^{10} + \frac{9}{95} a^{9} + \frac{1}{95} a^{8} - \frac{16}{95} a^{7} + \frac{1}{95} a^{6} - \frac{33}{190} a^{5} + \frac{91}{190} a^{4} + \frac{5}{38} a^{3} - \frac{46}{95} a^{2} + \frac{37}{190} a - \frac{93}{190}$, $\frac{1}{190} a^{13} + \frac{4}{95} a^{10} - \frac{6}{95} a^{9} - \frac{9}{190} a^{8} - \frac{77}{190} a^{7} + \frac{11}{190} a^{6} + \frac{43}{95} a^{5} - \frac{3}{38} a^{4} + \frac{41}{190} a^{3} + \frac{39}{95} a^{2} + \frac{43}{95} a + \frac{77}{190}$, $\frac{1}{570} a^{14} + \frac{1}{570} a^{13} - \frac{1}{570} a^{12} - \frac{1}{114} a^{10} - \frac{1}{114} a^{9} - \frac{1}{15} a^{8} - \frac{88}{285} a^{7} - \frac{13}{190} a^{6} + \frac{7}{57} a^{5} + \frac{271}{570} a^{4} + \frac{22}{95} a^{3} + \frac{28}{285} a^{2} + \frac{94}{285} a + \frac{23}{57}$, $\frac{1}{31017649528058194832756393045039622111459942475939350} a^{15} + \frac{12126539515665441371314989745237880672963987605441}{31017649528058194832756393045039622111459942475939350} a^{14} - \frac{5292651268192318396593582281264977911846320632607}{6203529905611638966551278609007924422291988495187870} a^{13} + \frac{12725272514369845920476418503761089111955313630337}{10339216509352731610918797681679874037153314158646450} a^{12} + \frac{12659415460578474428161387984274351448564432196367}{6203529905611638966551278609007924422291988495187870} a^{11} - \frac{765810187914131887057760706309179336687769302392031}{31017649528058194832756393045039622111459942475939350} a^{10} + \frac{1505550816877192219319971771388650602713688970090813}{31017649528058194832756393045039622111459942475939350} a^{9} - \frac{2887078228829961117701642536055430635655611833102589}{31017649528058194832756393045039622111459942475939350} a^{8} - \frac{659225751599938072720756629071862676657964613390789}{10339216509352731610918797681679874037153314158646450} a^{7} + \frac{11072672656788157473529032258832693834729270090796581}{31017649528058194832756393045039622111459942475939350} a^{6} + \frac{12159452413078108124864558170841876859540645339411247}{31017649528058194832756393045039622111459942475939350} a^{5} + \frac{589240558552103402849306743675941479958511393402769}{2067843301870546322183759536335974807430662831729290} a^{4} + \frac{3170189383151721836835482547416525283666046544743883}{31017649528058194832756393045039622111459942475939350} a^{3} - \frac{2789909021254002875687901574941954883155523905818501}{6203529905611638966551278609007924422291988495187870} a^{2} - \frac{15158203227113995405347462859383869677448208284579077}{31017649528058194832756393045039622111459942475939350} a - \frac{628868115610441897820517721445317777330098037258728}{5169608254676365805459398840839937018576657079323225}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}\times C_{4}\times C_{40}$, which has order $2560$ (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:  \( 6501070.6915 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4:C_4$ (as 16T8):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_4:C_4$
Character table for $C_4:C_4$

Intermediate fields

\(\Q(\sqrt{101}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{505}) \), 4.0.1030301.1, \(\Q(\sqrt{5}, \sqrt{101})\), 4.0.25757525.2, 4.0.12625.1 x2, 4.0.1275125.1 x2, 8.0.663450094125625.6, 8.0.1625943765625.5, 8.8.16586252353140625.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
$101$101.4.3.2$x^{4} - 404$$4$$1$$3$$C_4$$[\ ]_{4}$
101.4.3.2$x^{4} - 404$$4$$1$$3$$C_4$$[\ ]_{4}$
101.4.3.2$x^{4} - 404$$4$$1$$3$$C_4$$[\ ]_{4}$
101.4.3.2$x^{4} - 404$$4$$1$$3$$C_4$$[\ ]_{4}$