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Properties

Label 16.0.28621836185...0801.9
Degree $16$
Signature $[0, 8]$
Discriminant $17^{12}\cdot 53^{12}$
Root discriminant $164.45$
Ramified primes $17, 53$
Class number $19600$ (GRH)
Class group $[2, 70, 140]$ (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![90915613, -46191701, 68885217, -21086696, 15449301, -1990836, 1100675, -103927, 51943, -15529, 7486, -1058, 507, -42, 17, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 17*x^14 - 42*x^13 + 507*x^12 - 1058*x^11 + 7486*x^10 - 15529*x^9 + 51943*x^8 - 103927*x^7 + 1100675*x^6 - 1990836*x^5 + 15449301*x^4 - 21086696*x^3 + 68885217*x^2 - 46191701*x + 90915613)
 
gp: K = bnfinit(x^16 - 4*x^15 + 17*x^14 - 42*x^13 + 507*x^12 - 1058*x^11 + 7486*x^10 - 15529*x^9 + 51943*x^8 - 103927*x^7 + 1100675*x^6 - 1990836*x^5 + 15449301*x^4 - 21086696*x^3 + 68885217*x^2 - 46191701*x + 90915613, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 17 x^{14} - 42 x^{13} + 507 x^{12} - 1058 x^{11} + 7486 x^{10} - 15529 x^{9} + 51943 x^{8} - 103927 x^{7} + 1100675 x^{6} - 1990836 x^{5} + 15449301 x^{4} - 21086696 x^{3} + 68885217 x^{2} - 46191701 x + 90915613 \)

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:  \(286218361857094751614277009933470801=17^{12}\cdot 53^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $164.45$
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:  $17, 53$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2582976} a^{12} - \frac{1}{860992} a^{11} - \frac{6161}{1291488} a^{10} - \frac{108761}{1291488} a^{9} - \frac{99177}{430496} a^{8} - \frac{768137}{2582976} a^{7} + \frac{1096387}{2582976} a^{6} + \frac{3451}{80718} a^{5} + \frac{289457}{860992} a^{4} + \frac{130433}{860992} a^{3} + \frac{176785}{860992} a^{2} - \frac{440179}{2582976} a - \frac{498787}{2582976}$, $\frac{1}{2582976} a^{13} - \frac{1121}{234816} a^{11} - \frac{31811}{322872} a^{10} + \frac{3655}{215248} a^{9} + \frac{29653}{2582976} a^{8} - \frac{151003}{322872} a^{7} + \frac{816617}{2582976} a^{6} - \frac{30607}{860992} a^{5} + \frac{34453}{215248} a^{4} + \frac{3127}{19568} a^{3} - \frac{6391}{117408} a^{2} - \frac{131959}{645744} a + \frac{362205}{860992}$, $\frac{1}{2582976} a^{14} - \frac{291481}{2582976} a^{11} + \frac{248431}{1291488} a^{10} + \frac{62149}{860992} a^{9} + \frac{304379}{1291488} a^{8} + \frac{115377}{430496} a^{7} + \frac{39973}{645744} a^{6} - \frac{92353}{645744} a^{5} + \frac{17229}{78272} a^{4} + \frac{114517}{234816} a^{3} - \frac{815563}{2582976} a^{2} + \frac{35971}{1291488} a + \frac{74077}{234816}$, $\frac{1}{530401262967927876035929867836936384} a^{15} + \frac{1928840085341458023139225685}{530401262967927876035929867836936384} a^{14} + \frac{28842251923150636898919082849}{265200631483963938017964933918468192} a^{13} + \frac{401634995530632718488081483}{5525013155915915375374269456634754} a^{12} - \frac{7031230388932294593342073626053651}{33150078935495492252245616739808524} a^{11} + \frac{100271637046107110221189357445681731}{530401262967927876035929867836936384} a^{10} - \frac{10632832045843136874074106529796643}{176800420989309292011976622612312128} a^{9} + \frac{11403347327431906271463865968489467}{66300157870990984504491233479617048} a^{8} + \frac{62620607487926382986385615612994369}{530401262967927876035929867836936384} a^{7} - \frac{17981595992485995948527597589401825}{176800420989309292011976622612312128} a^{6} - \frac{59696635613665710115684856378332459}{176800420989309292011976622612312128} a^{5} + \frac{35751925462148950994863403715338987}{530401262967927876035929867836936384} a^{4} - \frac{2252051148106963599329291250859509}{16072765544482662910179692964755648} a^{3} + \frac{82969425769725857885205584735064227}{265200631483963938017964933918468192} a^{2} - \frac{442725157789467691938021797351159}{1880855542439460553318900240556512} a + \frac{2219993701127327245605368742724381}{88400210494654646005988311306156064}$

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

Class group and class number

$C_{2}\times C_{70}\times C_{140}$, which has order $19600$ (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:  \( 6596954.07823 \) (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

\(\Q(\sqrt{17}) \), \(\Q(\sqrt{53}) \), \(\Q(\sqrt{901}) \), \(\Q(\sqrt{17}, \sqrt{53})\), 4.0.2530909.1 x2, 4.0.43025453.1 x2, 4.4.13800617.2, 4.4.4913.1, 8.0.1851189605855209.3, 8.8.190457029580689.3, 8.0.534993796092155401.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{16}$ R ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$17$17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
17.4.3.1$x^{4} - 17$$4$$1$$3$$C_4$$[\ ]_{4}$
$53$53.8.6.1$x^{8} - 1643 x^{4} + 1755625$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
53.8.6.1$x^{8} - 1643 x^{4} + 1755625$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$