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Properties

Label 16.8.14014641336...5089.4
Degree $16$
Signature $[8, 4]$
Discriminant $37^{14}\cdot 41^{15}$
Root discriminant $765.88$
Ramified primes $37, 41$
Class number $8$ (GRH)
Class group $[8]$ (GRH)
Galois group $(C_2\times C_8).D_4$ (as 16T306)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8761246720, -29246158068, 25665275297, 8378734076, -7240728990, 3683185606, -1253263249, 1317824, 24934043, -16145710, 2532567, 379872, -63221, -70, 30, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 30*x^14 - 70*x^13 - 63221*x^12 + 379872*x^11 + 2532567*x^10 - 16145710*x^9 + 24934043*x^8 + 1317824*x^7 - 1253263249*x^6 + 3683185606*x^5 - 7240728990*x^4 + 8378734076*x^3 + 25665275297*x^2 - 29246158068*x + 8761246720)
 
gp: K = bnfinit(x^16 - 8*x^15 + 30*x^14 - 70*x^13 - 63221*x^12 + 379872*x^11 + 2532567*x^10 - 16145710*x^9 + 24934043*x^8 + 1317824*x^7 - 1253263249*x^6 + 3683185606*x^5 - 7240728990*x^4 + 8378734076*x^3 + 25665275297*x^2 - 29246158068*x + 8761246720, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 30 x^{14} - 70 x^{13} - 63221 x^{12} + 379872 x^{11} + 2532567 x^{10} - 16145710 x^{9} + 24934043 x^{8} + 1317824 x^{7} - 1253263249 x^{6} + 3683185606 x^{5} - 7240728990 x^{4} + 8378734076 x^{3} + 25665275297 x^{2} - 29246158068 x + 8761246720 \)

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:  \(14014641336002292789493279203934745080101945089=37^{14}\cdot 41^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $765.88$
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:  $37, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4} a$, $\frac{1}{20} a^{7} - \frac{1}{20} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{20} a^{3} - \frac{7}{20} a^{2} - \frac{1}{5} a$, $\frac{1}{1480} a^{8} - \frac{1}{370} a^{7} + \frac{59}{740} a^{6} - \frac{17}{74} a^{5} - \frac{81}{370} a^{4} - \frac{27}{148} a^{3} + \frac{567}{1480} a^{2} + \frac{63}{370} a + \frac{4}{37}$, $\frac{1}{1480} a^{9} + \frac{7}{370} a^{7} - \frac{41}{370} a^{6} - \frac{51}{370} a^{5} - \frac{43}{740} a^{4} - \frac{217}{1480} a^{3} + \frac{56}{185} a^{2} + \frac{177}{740} a + \frac{16}{37}$, $\frac{1}{14800} a^{10} - \frac{1}{2960} a^{9} + \frac{3}{14800} a^{8} + \frac{9}{7400} a^{7} + \frac{17}{7400} a^{6} - \frac{93}{7400} a^{5} - \frac{2787}{14800} a^{4} + \frac{1183}{2960} a^{3} - \frac{1229}{2960} a^{2} + \frac{197}{925} a - \frac{16}{185}$, $\frac{1}{14800} a^{11} - \frac{1}{7400} a^{9} + \frac{3}{14800} a^{8} + \frac{4}{925} a^{7} + \frac{163}{1850} a^{6} - \frac{1297}{14800} a^{5} + \frac{46}{185} a^{4} + \frac{85}{296} a^{3} + \frac{6557}{14800} a^{2} + \frac{27}{185} a + \frac{4}{37}$, $\frac{1}{118400} a^{12} + \frac{1}{59200} a^{11} + \frac{3}{118400} a^{10} + \frac{3}{14800} a^{9} + \frac{1}{23680} a^{8} + \frac{141}{59200} a^{7} - \frac{13199}{118400} a^{6} + \frac{589}{29600} a^{5} + \frac{2379}{23680} a^{4} - \frac{25589}{59200} a^{3} + \frac{56309}{118400} a^{2} + \frac{443}{5920} a + \frac{5}{37}$, $\frac{1}{592000} a^{13} + \frac{1}{592000} a^{12} + \frac{17}{592000} a^{11} + \frac{13}{592000} a^{10} - \frac{11}{592000} a^{9} - \frac{99}{592000} a^{8} + \frac{839}{592000} a^{7} + \frac{36307}{592000} a^{6} + \frac{2683}{23680} a^{5} + \frac{29303}{592000} a^{4} - \frac{47993}{592000} a^{3} + \frac{152143}{592000} a^{2} - \frac{18119}{148000} a + \frac{301}{925}$, $\frac{1}{331505023572860529560320000} a^{14} - \frac{7}{331505023572860529560320000} a^{13} + \frac{171530740123528240927}{165752511786430264780160000} a^{12} - \frac{2058368881482338891033}{331505023572860529560320000} a^{11} + \frac{160012772468927478741}{16575251178643026478016000} a^{10} + \frac{2867104166695358626869}{331505023572860529560320000} a^{9} - \frac{313509616673624345543}{41438127946607566195040000} a^{8} - \frac{5962474992584380100983}{66301004714572105912064000} a^{7} - \frac{130875415644857634788131}{1294941498331486443595000} a^{6} - \frac{65101165363612948377074277}{331505023572860529560320000} a^{5} + \frac{27844996044256109066677029}{165752511786430264780160000} a^{4} + \frac{52320959016502280156725677}{331505023572860529560320000} a^{3} + \frac{31503602043655005452016337}{66301004714572105912064000} a^{2} + \frac{41152325604487244790617787}{82876255893215132390080000} a + \frac{28607292812502122277549}{64747074916574322179750}$, $\frac{1}{505545160948612307579488000000} a^{15} + \frac{151}{101109032189722461515897600000} a^{14} + \frac{4138389964117403448243}{12638629023715307689487200000} a^{13} + \frac{264101307864223420974183}{101109032189722461515897600000} a^{12} - \frac{453653278706136638107163}{252772580474306153789744000000} a^{11} - \frac{2175529661241706396145491}{505545160948612307579488000000} a^{10} + \frac{4796745239404717805612717}{252772580474306153789744000000} a^{9} + \frac{100106022817335607677620557}{505545160948612307579488000000} a^{8} + \frac{3312111930455999417763116017}{252772580474306153789744000000} a^{7} + \frac{3942138942035052575970722891}{505545160948612307579488000000} a^{6} - \frac{8609593669018487509749550677}{63193145118576538447436000000} a^{5} + \frac{90085577815433597614293359073}{505545160948612307579488000000} a^{4} - \frac{66971376986727151625276876041}{505545160948612307579488000000} a^{3} + \frac{8938191134625478793389166959}{252772580474306153789744000000} a^{2} - \frac{9042490203925327634066388393}{63193145118576538447436000000} a + \frac{14878873779611461724745344}{49369644623887920662059375}$

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

Class group and class number

$C_{8}$, which has order $8$ (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:  \( 2479683760740000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times C_8).D_4$ (as 16T306):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $(C_2\times C_8).D_4$
Character table for $(C_2\times C_8).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{41}) \), 4.4.94352849.1, 8.8.499686183762100623329.2

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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{4}$ $16$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{4}$ $16$ $16$ $16$ $16$ $16$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ $16$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ R R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ $16$ $16$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$37$37.8.7.2$x^{8} - 148$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
37.8.7.2$x^{8} - 148$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
41Data not computed