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Properties

Label 16.0.29233126977...0625.6
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 149^{12}$
Root discriminant $142.60$
Ramified primes $5, 149$
Class number $18000$ (GRH)
Class group $[5, 60, 60]$ (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![45464483611, 174773137, 9069376118, -209534153, 1175303819, -92670513, 86452747, -9941937, 5325668, -596057, 227457, -19774, 6330, -364, 109, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 109*x^14 - 364*x^13 + 6330*x^12 - 19774*x^11 + 227457*x^10 - 596057*x^9 + 5325668*x^8 - 9941937*x^7 + 86452747*x^6 - 92670513*x^5 + 1175303819*x^4 - 209534153*x^3 + 9069376118*x^2 + 174773137*x + 45464483611)
 
gp: K = bnfinit(x^16 - 4*x^15 + 109*x^14 - 364*x^13 + 6330*x^12 - 19774*x^11 + 227457*x^10 - 596057*x^9 + 5325668*x^8 - 9941937*x^7 + 86452747*x^6 - 92670513*x^5 + 1175303819*x^4 - 209534153*x^3 + 9069376118*x^2 + 174773137*x + 45464483611, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 109 x^{14} - 364 x^{13} + 6330 x^{12} - 19774 x^{11} + 227457 x^{10} - 596057 x^{9} + 5325668 x^{8} - 9941937 x^{7} + 86452747 x^{6} - 92670513 x^{5} + 1175303819 x^{4} - 209534153 x^{3} + 9069376118 x^{2} + 174773137 x + 45464483611 \)

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:  \(29233126977165462000179004150390625=5^{12}\cdot 149^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $142.60$
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:  $5, 149$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{443841520} a^{12} - \frac{3}{443841520} a^{11} + \frac{11399765}{88768304} a^{10} - \frac{11963817}{88768304} a^{9} - \frac{8334177}{221920760} a^{8} + \frac{30285635}{88768304} a^{7} + \frac{94050243}{221920760} a^{6} - \frac{34908307}{88768304} a^{5} + \frac{5777123}{221920760} a^{4} - \frac{22199161}{88768304} a^{3} + \frac{12891577}{88768304} a^{2} + \frac{92369639}{221920760} a + \frac{11466631}{443841520}$, $\frac{1}{2219207600} a^{13} - \frac{1}{1109603800} a^{12} - \frac{10180071}{58400200} a^{11} + \frac{22051063}{110960380} a^{10} + \frac{145433321}{2219207600} a^{9} - \frac{309081699}{2219207600} a^{8} + \frac{339528661}{2219207600} a^{7} - \frac{22646451}{116800400} a^{6} + \frac{502774991}{2219207600} a^{5} + \frac{788241481}{2219207600} a^{4} - \frac{3561501}{11096038} a^{3} + \frac{914959443}{2219207600} a^{2} + \frac{14418161}{76524400} a + \frac{11466631}{2219207600}$, $\frac{1}{2219207600} a^{14} - \frac{1}{1109603800} a^{12} - \frac{95897109}{554801900} a^{11} - \frac{292700659}{2219207600} a^{10} + \frac{2796497}{116800400} a^{9} - \frac{496728737}{2219207600} a^{8} + \frac{325117853}{2219207600} a^{7} - \frac{1104819547}{2219207600} a^{6} - \frac{1066471637}{2219207600} a^{5} - \frac{297087119}{1109603800} a^{4} + \frac{304245343}{2219207600} a^{3} + \frac{42234331}{443841520} a^{2} + \frac{576436369}{2219207600} a + \frac{204347181}{1109603800}$, $\frac{1}{24696134648643879131491060828108753648972400} a^{15} - \frac{872821011945268338282287690128407}{24696134648643879131491060828108753648972400} a^{14} + \frac{431091272516945762071225915351429}{3087016831080484891436382603513594206121550} a^{13} + \frac{118836300627429446254655496319879}{129979656045494100692058214884782913941960} a^{12} - \frac{2443686072958757514251841290509464692243639}{24696134648643879131491060828108753648972400} a^{11} + \frac{1101688773891777406486338648541322268663049}{6174033662160969782872765207027188412243100} a^{10} + \frac{317283760331929839994812648537931619491247}{3087016831080484891436382603513594206121550} a^{9} + \frac{209535437395056223671689984041699823719423}{12348067324321939565745530414054376824486200} a^{8} - \frac{87232527820450590560043802240058795657743}{3087016831080484891436382603513594206121550} a^{7} + \frac{1133747957215738059401629251452593618470423}{12348067324321939565745530414054376824486200} a^{6} + \frac{2261921938146359927904490902103249807789763}{4939226929728775826298212165621750729794480} a^{5} + \frac{4712231189448563399708532309453909575085763}{24696134648643879131491060828108753648972400} a^{4} - \frac{1995502659699223747381372947191362836655823}{12348067324321939565745530414054376824486200} a^{3} + \frac{1770011222326214197526116508101546048757473}{12348067324321939565745530414054376824486200} a^{2} - \frac{469749435680895467202904663512679318523371}{987845385945755165259642433124350145958896} a + \frac{1632644392820247833192993350471941636989}{42579542497661860571536311772601299394780}$

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

Class group and class number

$C_{5}\times C_{60}\times C_{60}$, which has order $18000$ (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:  \( 2895687.10401 \) (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

\(\Q(\sqrt{149}) \), \(\Q(\sqrt{745}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{5}, \sqrt{149})\), 4.4.2775125.1 x2, 4.4.18625.1 x2, 4.0.3307949.1, 4.0.82698725.1, 8.8.7701318765625.1, 8.0.6839079116625625.1, 8.0.170976977915640625.1

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
$149$149.8.6.1$x^{8} - 1043 x^{4} + 1798281$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
149.8.6.1$x^{8} - 1043 x^{4} + 1798281$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$