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Properties

Label 16.0.42832588164...5409.3
Degree $16$
Signature $[0, 8]$
Discriminant $3^{8}\cdot 97^{14}$
Root discriminant $94.84$
Ramified primes $3, 97$
Class number $136$ (GRH)
Class group $[136]$ (GRH)
Galois group $C_8: C_2$ (as 16T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16589329, -3295057, 5582811, 6835864, -1141852, 1914443, 823930, -184658, 236155, -30672, 21216, -2203, 1326, -76, 43, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 43*x^14 - 76*x^13 + 1326*x^12 - 2203*x^11 + 21216*x^10 - 30672*x^9 + 236155*x^8 - 184658*x^7 + 823930*x^6 + 1914443*x^5 - 1141852*x^4 + 6835864*x^3 + 5582811*x^2 - 3295057*x + 16589329)
 
gp: K = bnfinit(x^16 - x^15 + 43*x^14 - 76*x^13 + 1326*x^12 - 2203*x^11 + 21216*x^10 - 30672*x^9 + 236155*x^8 - 184658*x^7 + 823930*x^6 + 1914443*x^5 - 1141852*x^4 + 6835864*x^3 + 5582811*x^2 - 3295057*x + 16589329, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 43 x^{14} - 76 x^{13} + 1326 x^{12} - 2203 x^{11} + 21216 x^{10} - 30672 x^{9} + 236155 x^{8} - 184658 x^{7} + 823930 x^{6} + 1914443 x^{5} - 1141852 x^{4} + 6835864 x^{3} + 5582811 x^{2} - 3295057 x + 16589329 \)

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:  \(42832588164190465585875603485409=3^{8}\cdot 97^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $94.84$
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:  $3, 97$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{6} a^{12} + \frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{1}{6} a^{9} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{6} a^{6} + \frac{1}{3} a^{4} - \frac{1}{2} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{6}$, $\frac{1}{6} a^{13} - \frac{1}{3} a^{11} - \frac{1}{2} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{2} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{6} a^{4} - \frac{1}{3} a^{3} - \frac{1}{2} a - \frac{1}{3}$, $\frac{1}{16401318} a^{14} + \frac{20281}{381426} a^{13} - \frac{872041}{16401318} a^{12} + \frac{2268683}{5467106} a^{11} + \frac{4640047}{16401318} a^{10} + \frac{1354097}{16401318} a^{9} - \frac{24439}{5467106} a^{8} - \frac{1733903}{5467106} a^{7} + \frac{995465}{5467106} a^{6} + \frac{1545319}{16401318} a^{5} + \frac{3846125}{16401318} a^{4} - \frac{5593223}{16401318} a^{3} - \frac{7423409}{16401318} a^{2} - \frac{928397}{5467106} a + \frac{5776205}{16401318}$, $\frac{1}{77875559286888959565243643393322138267682} a^{15} - \frac{827690106968604501718387166368916}{38937779643444479782621821696661069133841} a^{14} + \frac{392275928496152959411858485856704241790}{38937779643444479782621821696661069133841} a^{13} + \frac{746895064364845174499686413478903738133}{38937779643444479782621821696661069133841} a^{12} + \frac{1578100865096469648632919100263490911839}{12979259881148159927540607232220356377947} a^{11} + \frac{12261395825962257984647375343471371999}{905529759149871622851670272015373700787} a^{10} - \frac{2088705144595343094084607261587574528569}{12979259881148159927540607232220356377947} a^{9} + \frac{897645120927268541736045396406337024417}{38937779643444479782621821696661069133841} a^{8} - \frac{84443196594020808595508910916745897483}{257866090353936952202793521169940855191} a^{7} + \frac{1486781967371696103974471259666692720753}{38937779643444479782621821696661069133841} a^{6} - \frac{11588582620062965047522269573163354095863}{38937779643444479782621821696661069133841} a^{5} + \frac{2266136645758247334179758763937541655295}{12979259881148159927540607232220356377947} a^{4} + \frac{12929923909485313454112901143204690061}{85955363451312317400931173723313618397} a^{3} + \frac{1565785538307174645727822731618281487761}{12979259881148159927540607232220356377947} a^{2} - \frac{3135409564375008681749011923040006643642}{12979259881148159927540607232220356377947} a - \frac{6648566900944276219182929131806777737}{19119950721062843006443320253700500434}$

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

Class group and class number

$C_{136}$, which has order $136$ (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:  \( \frac{1674608685207860955048770747}{14244384375735345092128018625086497} a^{15} - \frac{5251761787129308073907972723}{9496256250490230061418679083390998} a^{14} + \frac{68442893877200314455233954251}{14244384375735345092128018625086497} a^{13} - \frac{127588873855443942380973621805}{4748128125245115030709339541695499} a^{12} + \frac{1535116561460614017215832356977}{9496256250490230061418679083390998} a^{11} - \frac{11121811007919877593504744253241}{14244384375735345092128018625086497} a^{10} + \frac{37423571675159992780016157170137}{14244384375735345092128018625086497} a^{9} - \frac{322287642366965360828663832045887}{28488768751470690184256037250172994} a^{8} + \frac{413721963919784540987461228920049}{14244384375735345092128018625086497} a^{7} - \frac{1488103602391888422610035908626063}{14244384375735345092128018625086497} a^{6} + \frac{489548323915829792622695660636589}{9496256250490230061418679083390998} a^{5} - \frac{326152388840937823365962666834723}{14244384375735345092128018625086497} a^{4} - \frac{16145974096710744872692276966792423}{14244384375735345092128018625086497} a^{3} - \frac{2427590921790621396606783103826969}{28488768751470690184256037250172994} a^{2} - \frac{2270338309686647187366079162180465}{4748128125245115030709339541695499} a - \frac{9848282116642884068433075004886}{3497270900008677901332683188089} \) (order $6$)
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:  \( 38023530.1922 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$OD_{16}$ (as 16T6):

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 $C_8: C_2$
Character table for $C_8: C_2$

Intermediate fields

\(\Q(\sqrt{-291}) \), \(\Q(\sqrt{97}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-3}, \sqrt{97})\), 4.4.912673.1, 4.0.8214057.1, 8.0.67470732399249.1, 8.4.727184560303017.1 x2

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

Sibling fields

Degree 8 sibling: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$97$97.8.7.3$x^{8} - 60625$$8$$1$$7$$C_8$$[\ ]_{8}$
97.8.7.3$x^{8} - 60625$$8$$1$$7$$C_8$$[\ ]_{8}$