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Properties

Label 16.8.19021991394...0000.2
Degree $16$
Signature $[8, 4]$
Discriminant $2^{44}\cdot 5^{14}\cdot 11^{6}$
Root discriminant $67.60$
Ramified primes $2, 5, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $(C_2^2\times C_4).C_2^4$ (as 16T471)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3025, 0, -7700, 0, -20050, 0, 35600, 0, 35695, 0, 4280, 0, -310, 0, -20, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 20*x^14 - 310*x^12 + 4280*x^10 + 35695*x^8 + 35600*x^6 - 20050*x^4 - 7700*x^2 + 3025)
 
gp: K = bnfinit(x^16 - 20*x^14 - 310*x^12 + 4280*x^10 + 35695*x^8 + 35600*x^6 - 20050*x^4 - 7700*x^2 + 3025, 1)
 

Normalized defining polynomial

\( x^{16} - 20 x^{14} - 310 x^{12} + 4280 x^{10} + 35695 x^{8} + 35600 x^{6} - 20050 x^{4} - 7700 x^{2} + 3025 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(190219913946726400000000000000=2^{44}\cdot 5^{14}\cdot 11^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $67.60$
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:  $2, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{40} a^{8} - \frac{1}{4} a^{6} + \frac{3}{8} a^{4} - \frac{1}{4} a^{2} - \frac{1}{8}$, $\frac{1}{40} a^{9} - \frac{1}{4} a^{7} + \frac{3}{8} a^{5} - \frac{1}{4} a^{3} - \frac{1}{8} a$, $\frac{1}{40} a^{10} - \frac{1}{8} a^{6} - \frac{1}{2} a^{4} + \frac{3}{8} a^{2} - \frac{1}{4}$, $\frac{1}{40} a^{11} - \frac{1}{8} a^{7} - \frac{1}{2} a^{5} + \frac{3}{8} a^{3} - \frac{1}{4} a$, $\frac{1}{760} a^{12} - \frac{3}{760} a^{10} - \frac{3}{760} a^{8} - \frac{53}{152} a^{6} + \frac{61}{152} a^{4} - \frac{63}{152} a^{2} - \frac{5}{38}$, $\frac{1}{760} a^{13} - \frac{3}{760} a^{11} - \frac{3}{760} a^{9} - \frac{53}{152} a^{7} + \frac{61}{152} a^{5} - \frac{63}{152} a^{3} - \frac{5}{38} a$, $\frac{1}{6882704717680} a^{14} - \frac{1}{1520} a^{13} - \frac{225991657}{344135235884} a^{12} + \frac{3}{1520} a^{11} - \frac{21726349889}{6882704717680} a^{10} - \frac{1}{95} a^{9} + \frac{59400442421}{6882704717680} a^{8} + \frac{91}{304} a^{7} + \frac{60549080495}{125140085776} a^{6} + \frac{17}{152} a^{5} - \frac{293257262471}{1376540943536} a^{4} - \frac{51}{304} a^{3} - \frac{85025100867}{688270471768} a^{2} - \frac{113}{304} a - \frac{39120733999}{125140085776}$, $\frac{1}{6882704717680} a^{15} + \frac{8262069}{6882704717680} a^{13} - \frac{1}{1520} a^{12} - \frac{8827658879}{1720676179420} a^{11} + \frac{3}{1520} a^{10} - \frac{40217652177}{6882704717680} a^{9} - \frac{1}{95} a^{8} + \frac{27187202605}{62570042888} a^{7} + \frac{91}{304} a^{6} + \frac{413125590133}{1376540943536} a^{5} + \frac{17}{152} a^{4} + \frac{405017889809}{1376540943536} a^{3} - \frac{51}{304} a^{2} + \frac{11518831935}{62570042888} a - \frac{113}{304}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $11$
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:  \( 819805611.743 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2^2\times C_4).C_2^4$ (as 16T471):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 256
The 46 conjugacy class representatives for $(C_2^2\times C_4).C_2^4$
Character table for $(C_2^2\times C_4).C_2^4$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.88000.1, \(\Q(\zeta_{20})^+\), 4.4.17600.1, 8.8.123904000000.4

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
2Data not computed
5Data not computed
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$