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Properties

Label 16.0.67018809606...1721.1
Degree $16$
Signature $[0, 8]$
Discriminant $29^{14}\cdot 83^{8}$
Root discriminant $173.44$
Ramified primes $29, 83$
Class number $30$ (GRH)
Class group $[30]$ (GRH)
Galois group $C_2^3.C_4$ (as 16T36)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![31088575, -64246320, 106278189, 76161103, -11440997, -31438382, 7200689, -556257, 284435, 35819, 6979, -1754, 848, -154, 49, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 49*x^14 - 154*x^13 + 848*x^12 - 1754*x^11 + 6979*x^10 + 35819*x^9 + 284435*x^8 - 556257*x^7 + 7200689*x^6 - 31438382*x^5 - 11440997*x^4 + 76161103*x^3 + 106278189*x^2 - 64246320*x + 31088575)
 
gp: K = bnfinit(x^16 - 4*x^15 + 49*x^14 - 154*x^13 + 848*x^12 - 1754*x^11 + 6979*x^10 + 35819*x^9 + 284435*x^8 - 556257*x^7 + 7200689*x^6 - 31438382*x^5 - 11440997*x^4 + 76161103*x^3 + 106278189*x^2 - 64246320*x + 31088575, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 49 x^{14} - 154 x^{13} + 848 x^{12} - 1754 x^{11} + 6979 x^{10} + 35819 x^{9} + 284435 x^{8} - 556257 x^{7} + 7200689 x^{6} - 31438382 x^{5} - 11440997 x^{4} + 76161103 x^{3} + 106278189 x^{2} - 64246320 x + 31088575 \)

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:  \(670188096064724500217499812793861721=29^{14}\cdot 83^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $173.44$
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:  $29, 83$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{7} a^{7} - \frac{1}{7} a$, $\frac{1}{7} a^{8} - \frac{1}{7} a^{2}$, $\frac{1}{7} a^{9} - \frac{1}{7} a^{3}$, $\frac{1}{7} a^{10} - \frac{1}{7} a^{4}$, $\frac{1}{7} a^{11} - \frac{1}{7} a^{5}$, $\frac{1}{49} a^{12} + \frac{3}{49} a^{11} + \frac{2}{49} a^{10} - \frac{1}{49} a^{9} - \frac{3}{49} a^{8} - \frac{2}{49} a^{7} + \frac{6}{49} a^{6} - \frac{24}{49} a^{5} - \frac{16}{49} a^{4} + \frac{8}{49} a^{3} + \frac{24}{49} a^{2} + \frac{16}{49} a - \frac{2}{7}$, $\frac{1}{245} a^{13} + \frac{2}{245} a^{12} - \frac{1}{245} a^{11} + \frac{11}{245} a^{10} - \frac{9}{245} a^{9} + \frac{3}{49} a^{8} - \frac{13}{245} a^{7} - \frac{79}{245} a^{6} + \frac{57}{245} a^{5} + \frac{108}{245} a^{4} + \frac{23}{245} a^{3} - \frac{24}{49} a^{2} - \frac{58}{245} a - \frac{1}{7}$, $\frac{1}{121755902708755} a^{14} - \frac{119615023182}{121755902708755} a^{13} + \frac{1119061807081}{121755902708755} a^{12} - \frac{1493886611362}{24351180541751} a^{11} + \frac{13493425973}{239206095695} a^{10} + \frac{3284043991066}{121755902708755} a^{9} - \frac{4577724990648}{121755902708755} a^{8} - \frac{5165636889477}{121755902708755} a^{7} - \frac{60843182233072}{121755902708755} a^{6} + \frac{4720472585841}{24351180541751} a^{5} + \frac{43520087800236}{121755902708755} a^{4} - \frac{51642933771197}{121755902708755} a^{3} - \frac{41037158909078}{121755902708755} a^{2} - \frac{38839989511178}{121755902708755} a - \frac{21073538876}{58961696227}$, $\frac{1}{9163680055032722060121088396009151861408515} a^{15} + \frac{2852438776158780430021692616}{9163680055032722060121088396009151861408515} a^{14} + \frac{13011792920865386984168678672914076233627}{9163680055032722060121088396009151861408515} a^{13} + \frac{2347339283414619313029900866923373595222}{9163680055032722060121088396009151861408515} a^{12} - \frac{26033659101502127451407240373168726145805}{1832736011006544412024217679201830372281703} a^{11} + \frac{459647897383097657265473726555404312538314}{9163680055032722060121088396009151861408515} a^{10} - \frac{107122864075687083757581981985174432651638}{9163680055032722060121088396009151861408515} a^{9} + \frac{12906824552198930657280090120587603019064}{9163680055032722060121088396009151861408515} a^{8} + \frac{16253422198756386967610018898726208024273}{1309097150718960294303012628001307408772645} a^{7} - \frac{980472966818026042363875237327967636638694}{9163680055032722060121088396009151861408515} a^{6} + \frac{31512054622407400024876611743200354703712}{79684174391588887479313812139210016186161} a^{5} + \frac{1075701930694478097454361197629987607398397}{9163680055032722060121088396009151861408515} a^{4} - \frac{3798653094962534665990067894826993467818768}{9163680055032722060121088396009151861408515} a^{3} + \frac{3693972072300446684273783000583063105659338}{9163680055032722060121088396009151861408515} a^{2} + \frac{218160823537157055911432571096107307331816}{1832736011006544412024217679201830372281703} a + \frac{257868446976645375276960325144131454008}{4437617460064272184078008908479008165331}$

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

Class group and class number

$C_{30}$, which has order $30$ (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:  \( 89005085005.7 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_4$ (as 16T36):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $C_2^3.C_4$
Character table for $C_2^3.C_4$

Intermediate fields

\(\Q(\sqrt{-83}) \), \(\Q(\sqrt{29}) \), \(\Q(\sqrt{-2407}) \), 4.4.168015821.1, 4.0.24389.1, \(\Q(\sqrt{29}, \sqrt{-83})\), 8.4.818650167082817189.1 x2, 8.0.118834397892701.1 x2, 8.0.28229316106304041.1

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

Sibling fields

Galois closure: data not computed
Degree 8 siblings: data not computed
Degree 16 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/7.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ R ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{16}$

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
$29$29.8.7.2$x^{8} - 116$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
29.8.7.2$x^{8} - 116$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$83$83.2.1.2$x^{2} + 249$$2$$1$$1$$C_2$$[\ ]_{2}$
83.2.1.2$x^{2} + 249$$2$$1$$1$$C_2$$[\ ]_{2}$
83.2.1.2$x^{2} + 249$$2$$1$$1$$C_2$$[\ ]_{2}$
83.2.1.2$x^{2} + 249$$2$$1$$1$$C_2$$[\ ]_{2}$
83.4.2.1$x^{4} + 249 x^{2} + 27556$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
83.4.2.1$x^{4} + 249 x^{2} + 27556$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$