↑

Properties

Label 16.0.12476445670...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{8}\cdot 5^{14}\cdot 41^{8}$
Root discriminant $37.03$
Ramified primes $2, 5, 41$
Class number $8$ (GRH)
Class group $[2, 2, 2]$ (GRH)
Galois group $(C_2\times Q_8).C_2^3$ (as 16T226)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![274576, 129952, 322952, 116616, 182396, 39372, 67910, 7188, 16899, 419, 3195, -166, 406, -22, 28, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 + 28*x^14 - 22*x^13 + 406*x^12 - 166*x^11 + 3195*x^10 + 419*x^9 + 16899*x^8 + 7188*x^7 + 67910*x^6 + 39372*x^5 + 182396*x^4 + 116616*x^3 + 322952*x^2 + 129952*x + 274576)
 
gp: K = bnfinit(x^16 - x^15 + 28*x^14 - 22*x^13 + 406*x^12 - 166*x^11 + 3195*x^10 + 419*x^9 + 16899*x^8 + 7188*x^7 + 67910*x^6 + 39372*x^5 + 182396*x^4 + 116616*x^3 + 322952*x^2 + 129952*x + 274576, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} + 28 x^{14} - 22 x^{13} + 406 x^{12} - 166 x^{11} + 3195 x^{10} + 419 x^{9} + 16899 x^{8} + 7188 x^{7} + 67910 x^{6} + 39372 x^{5} + 182396 x^{4} + 116616 x^{3} + 322952 x^{2} + 129952 x + 274576 \)

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:  \(12476445670501562500000000=2^{8}\cdot 5^{14}\cdot 41^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.03$
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:  $2, 5, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{2} a^{9} + \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4}$, $\frac{1}{8} a^{14} - \frac{1}{8} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{10} + \frac{1}{4} a^{9} + \frac{3}{8} a^{8} + \frac{3}{8} a^{7} - \frac{1}{8} a^{6} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{10762952189662400129628043467843688} a^{15} + \frac{492881796929738572629752899170175}{10762952189662400129628043467843688} a^{14} + \frac{192175068074366791385380887842131}{5381476094831200064814021733921844} a^{13} - \frac{261673188953024557691985877948851}{2690738047415600032407010866960922} a^{12} - \frac{1009384300207828561276914284838847}{5381476094831200064814021733921844} a^{11} + \frac{65838124200419678851612550969885}{489225099530109096801274703083804} a^{10} - \frac{4286729829237956340378678484687001}{10762952189662400129628043467843688} a^{9} - \frac{222596457852024155965718363603665}{10762952189662400129628043467843688} a^{8} + \frac{24487269864972143560188655971843}{82159940379102291065862927235448} a^{7} + \frac{1821743310098278314472449358586327}{5381476094831200064814021733921844} a^{6} + \frac{424531645468826114862506911174585}{1345369023707800016203505433480461} a^{5} - \frac{28403284602241376685134719462465}{122306274882527274200318675770951} a^{4} - \frac{462988416949086555135717835914306}{1345369023707800016203505433480461} a^{3} - \frac{281566710594469754029056617148852}{1345369023707800016203505433480461} a^{2} - \frac{50213856972152988426813564137484}{122306274882527274200318675770951} a - \frac{3183110600974255247012745542159}{10269992547387786383232865904431}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}$, 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:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{11295486078016229085505169}{587433259996856245476915373204} a^{15} + \frac{71300246747195958576648077}{1174866519993712490953830746408} a^{14} - \frac{599897830267985283087543779}{1174866519993712490953830746408} a^{13} + \frac{405418607519473332959383577}{293716629998428122738457686602} a^{12} - \frac{4093054663357615140602711983}{587433259996856245476915373204} a^{11} + \frac{806704767148946403652601119}{53403023636077840497901397564} a^{10} - \frac{6535391070261575692289332833}{146858314999214061369228843301} a^{9} + \frac{79268753620244658669261364979}{1174866519993712490953830746408} a^{8} - \frac{1381925724055399297743235463}{8968446717509255656136112568} a^{7} + \frac{295430138311083549617736772469}{1174866519993712490953830746408} a^{6} - \frac{61283912434128802283308869011}{146858314999214061369228843301} a^{5} + \frac{36255238348066181354571321931}{53403023636077840497901397564} a^{4} - \frac{56806180397489836176084086166}{146858314999214061369228843301} a^{3} + \frac{333055755901800338331803638281}{293716629998428122738457686602} a^{2} - \frac{1825439871545751028595179395}{13350755909019460124475349391} a + \frac{1851024983051557478786199463}{1121055839688656957017014071} \) (order $10$)
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:  \( 622261.644536 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times Q_8).C_2^3$ (as 16T226):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 23 conjugacy class representatives for $(C_2\times Q_8).C_2^3$
Character table for $(C_2\times Q_8).C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 4.0.1025.1, 4.4.5125.1, 8.0.26265625.1

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

Sibling fields

Degree 16 siblings: 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 R ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.8.8.4$x^{8} + 2 x^{7} + 2 x^{6} + 8 x^{3} + 48$$2$$4$$8$$C_8$$[2]^{4}$
2.8.0.1$x^{8} + x^{4} + x^{3} + x + 1$$1$$8$$0$$C_8$$[\ ]^{8}$
$5$5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
5.8.7.2$x^{8} - 20$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$41$41.4.2.2$x^{4} - 41 x^{2} + 20172$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
41.4.3.4$x^{4} + 8856$$4$$1$$3$$C_4$$[\ ]_{4}$
41.4.0.1$x^{4} - x + 17$$1$$4$$0$$C_4$$[\ ]^{4}$
41.4.3.3$x^{4} + 246$$4$$1$$3$$C_4$$[\ ]_{4}$