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Properties

Label 16.0.37038789304...0625.3
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 79^{8}$
Root discriminant $29.72$
Ramified primes $5, 79$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2\wr C_4$ (as 16T172)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4096, 512, -1664, 2400, 6113, -126, -6593, 4169, 4237, -1963, -567, 80, 95, -12, 0, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - 12*x^13 + 95*x^12 + 80*x^11 - 567*x^10 - 1963*x^9 + 4237*x^8 + 4169*x^7 - 6593*x^6 - 126*x^5 + 6113*x^4 + 2400*x^3 - 1664*x^2 + 512*x + 4096)
 
gp: K = bnfinit(x^16 - 3*x^15 - 12*x^13 + 95*x^12 + 80*x^11 - 567*x^10 - 1963*x^9 + 4237*x^8 + 4169*x^7 - 6593*x^6 - 126*x^5 + 6113*x^4 + 2400*x^3 - 1664*x^2 + 512*x + 4096, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} - 12 x^{13} + 95 x^{12} + 80 x^{11} - 567 x^{10} - 1963 x^{9} + 4237 x^{8} + 4169 x^{7} - 6593 x^{6} - 126 x^{5} + 6113 x^{4} + 2400 x^{3} - 1664 x^{2} + 512 x + 4096 \)

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:  \(370387893043593994140625=5^{12}\cdot 79^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.72$
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:  $5, 79$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{10} a^{12} + \frac{1}{10} a^{11} - \frac{1}{5} a^{10} - \frac{1}{5} a^{9} + \frac{1}{5} a^{8} + \frac{1}{10} a^{7} - \frac{1}{10} a^{6} + \frac{1}{5} a^{5} - \frac{1}{5} a^{4} + \frac{2}{5} a^{3} - \frac{1}{10} a^{2} + \frac{1}{10} a - \frac{2}{5}$, $\frac{1}{40} a^{13} + \frac{1}{40} a^{12} + \frac{1}{5} a^{11} + \frac{1}{5} a^{10} + \frac{7}{40} a^{9} - \frac{1}{10} a^{8} + \frac{9}{40} a^{7} - \frac{3}{40} a^{6} + \frac{13}{40} a^{5} - \frac{11}{40} a^{4} + \frac{19}{40} a^{3} + \frac{3}{20} a^{2} - \frac{19}{40} a$, $\frac{1}{9920} a^{14} + \frac{109}{9920} a^{13} + \frac{5}{124} a^{12} - \frac{571}{2480} a^{11} - \frac{1057}{9920} a^{10} + \frac{11}{155} a^{9} + \frac{917}{1984} a^{8} + \frac{4773}{9920} a^{7} - \frac{423}{1984} a^{6} - \frac{679}{9920} a^{5} + \frac{3983}{9920} a^{4} + \frac{1417}{4960} a^{3} + \frac{909}{1984} a^{2} + \frac{237}{620} a + \frac{51}{155}$, $\frac{1}{3186870591311480184287490560} a^{15} + \frac{19627631558419061411377}{637374118262296036857498112} a^{14} - \frac{87447299767185294795253}{398358823913935023035936320} a^{13} + \frac{25223748276339769225494413}{796717647827870046071872640} a^{12} - \frac{627936688528082590031558721}{3186870591311480184287490560} a^{11} - \frac{13251780807679266874274609}{79671764782787004607187264} a^{10} + \frac{155934541345535446804790325}{637374118262296036857498112} a^{9} + \frac{1045353528938431689726775053}{3186870591311480184287490560} a^{8} + \frac{11549319385786962914766497}{637374118262296036857498112} a^{7} + \frac{688678925770971467976348961}{3186870591311480184287490560} a^{6} + \frac{969349213375931093127909943}{3186870591311480184287490560} a^{5} - \frac{627746918933146445771317019}{1593435295655740092143745280} a^{4} - \frac{1185654735127399929080472687}{3186870591311480184287490560} a^{3} + \frac{14121339787603705318589087}{79671764782787004607187264} a^{2} + \frac{4361675445031960639784443}{49794852989241877879492040} a + \frac{301154229914406249199739}{6224356623655234734936505}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( -\frac{100639453581009463703137}{102802277139080005944757760} a^{15} + \frac{421346612338210509370747}{102802277139080005944757760} a^{14} - \frac{58928205800709690303521}{12850284642385000743094720} a^{13} + \frac{426297054167750012678051}{25700569284770001486189440} a^{12} - \frac{11611651572145943885475871}{102802277139080005944757760} a^{11} + \frac{131660476077817334014383}{2570056928477000148618944} a^{10} + \frac{53266664599291073532770391}{102802277139080005944757760} a^{9} + \frac{138300110135823535657299939}{102802277139080005944757760} a^{8} - \frac{600774136148411814531713333}{102802277139080005944757760} a^{7} + \frac{227635533702177452277987023}{102802277139080005944757760} a^{6} + \frac{447675528782405324273613113}{102802277139080005944757760} a^{5} - \frac{190709956090297498773064909}{51401138569540002972378880} a^{4} - \frac{96319239186646366174982801}{102802277139080005944757760} a^{3} - \frac{19633498213117386134788737}{12850284642385000743094720} a^{2} + \frac{2452703552591567820799641}{803142790149062546443420} a - \frac{534749741664954279342058}{200785697537265636610855} \) (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:  \( 361613.952106 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\wr C_4$ (as 16T172):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 13 conjugacy class representatives for $C_2\wr C_4$
Character table for $C_2\wr C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 4.2.1975.1, 4.2.9875.1, 8.4.121719003125.1, 8.0.121719003125.1, 8.0.97515625.1

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

Sibling fields

Degree 8 siblings: data not computed
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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$79$79.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
79.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
79.4.2.1$x^{4} + 395 x^{2} + 56169$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
79.8.6.1$x^{8} - 553 x^{4} + 505521$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$