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Properties

Label 16.0.83253845920...0816.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{26}\cdot 3^{14}\cdot 11^{10}$
Root discriminant $36.10$
Ramified primes $2, 3, 11$
Class number $40$ (GRH)
Class group $[2, 2, 10]$ (GRH)
Galois group $C_4^2:C_2^2.C_2$ (as 16T406)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![6196, 5880, -3604, -9336, 10306, -5748, 3590, 96, 1165, -126, 410, -54, 100, -12, 17, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 17*x^14 - 12*x^13 + 100*x^12 - 54*x^11 + 410*x^10 - 126*x^9 + 1165*x^8 + 96*x^7 + 3590*x^6 - 5748*x^5 + 10306*x^4 - 9336*x^3 - 3604*x^2 + 5880*x + 6196)
 
gp: K = bnfinit(x^16 + 17*x^14 - 12*x^13 + 100*x^12 - 54*x^11 + 410*x^10 - 126*x^9 + 1165*x^8 + 96*x^7 + 3590*x^6 - 5748*x^5 + 10306*x^4 - 9336*x^3 - 3604*x^2 + 5880*x + 6196, 1)
 

Normalized defining polynomial

\( x^{16} + 17 x^{14} - 12 x^{13} + 100 x^{12} - 54 x^{11} + 410 x^{10} - 126 x^{9} + 1165 x^{8} + 96 x^{7} + 3590 x^{6} - 5748 x^{5} + 10306 x^{4} - 9336 x^{3} - 3604 x^{2} + 5880 x + 6196 \)

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:  \(8325384592016962870050816=2^{26}\cdot 3^{14}\cdot 11^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.10$
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:  $2, 3, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{633350389759279475887499593692326} a^{15} - \frac{74908409049340673546908137794655}{316675194879639737943749796846163} a^{14} + \frac{62202415224986800950855428673087}{316675194879639737943749796846163} a^{13} + \frac{50071601817254928860367022607220}{316675194879639737943749796846163} a^{12} - \frac{258007946079185203571600566662219}{633350389759279475887499593692326} a^{11} - \frac{41159354422393659418678657035422}{316675194879639737943749796846163} a^{10} - \frac{107168469567326296128205002029156}{316675194879639737943749796846163} a^{9} - \frac{134402861096080123550724424405752}{316675194879639737943749796846163} a^{8} + \frac{7484922133764226012271039229263}{633350389759279475887499593692326} a^{7} + \frac{46978105621799107300877833493309}{316675194879639737943749796846163} a^{6} + \frac{33864433372507954845396355014987}{633350389759279475887499593692326} a^{5} - \frac{104768884294547978966225802759151}{316675194879639737943749796846163} a^{4} - \frac{13299922554285565837981784934199}{316675194879639737943749796846163} a^{3} - \frac{29372622131508635750560319663797}{316675194879639737943749796846163} a^{2} - \frac{52643566232655353915462734397122}{316675194879639737943749796846163} a + \frac{77258179486073388572934269603992}{316675194879639737943749796846163}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{10}$, which has order $40$ (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:  \( 15290.5686164 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4^2:C_2^2.C_2$ (as 16T406):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $C_4^2:C_2^2.C_2$
Character table for $C_4^2:C_2^2.C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{33}) \), \(\Q(\sqrt{11}) \), 4.4.4752.1 x2, 4.4.13068.1 x2, \(\Q(\sqrt{3}, \sqrt{11})\), 8.8.2732361984.1

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 R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$2$2.8.18.3$x^{8} + 14 x^{6} + 10 x^{4} + 8 x^{3} + 12 x^{2} + 20$$4$$2$$18$$Q_8:C_2$$[2, 3, 7/2]^{2}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
3Data not computed
$11$11.8.6.1$x^{8} + 143 x^{4} + 5929$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
11.8.4.1$x^{8} + 484 x^{4} - 1331 x^{2} + 58564$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$