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Properties

Label 16.0.51399544780...7056.4
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 3^{12}\cdot 7^{8}$
Root discriminant $17.06$
Ramified primes $2, 3, 7$
Class number $2$
Class group $[2]$
Galois group $C_2^2\wr C_2$ (as 16T39)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9, -72, 324, -900, 1866, -3156, 4356, -4860, 4411, -3260, 1972, -980, 406, -140, 40, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 40*x^14 - 140*x^13 + 406*x^12 - 980*x^11 + 1972*x^10 - 3260*x^9 + 4411*x^8 - 4860*x^7 + 4356*x^6 - 3156*x^5 + 1866*x^4 - 900*x^3 + 324*x^2 - 72*x + 9)
 
gp: K = bnfinit(x^16 - 8*x^15 + 40*x^14 - 140*x^13 + 406*x^12 - 980*x^11 + 1972*x^10 - 3260*x^9 + 4411*x^8 - 4860*x^7 + 4356*x^6 - 3156*x^5 + 1866*x^4 - 900*x^3 + 324*x^2 - 72*x + 9, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 40 x^{14} - 140 x^{13} + 406 x^{12} - 980 x^{11} + 1972 x^{10} - 3260 x^{9} + 4411 x^{8} - 4860 x^{7} + 4356 x^{6} - 3156 x^{5} + 1866 x^{4} - 900 x^{3} + 324 x^{2} - 72 x + 9 \)

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:  \(51399544780206637056=2^{24}\cdot 3^{12}\cdot 7^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.06$
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:  $2, 3, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{12} a^{8} - \frac{1}{3} a^{7} + \frac{1}{6} a^{5} - \frac{5}{12} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{12} a^{9} - \frac{1}{3} a^{7} + \frac{1}{6} a^{6} + \frac{1}{4} a^{5} - \frac{1}{6} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a$, $\frac{1}{12} a^{10} - \frac{1}{6} a^{7} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{6} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{12} a^{11} - \frac{5}{12} a^{7} - \frac{1}{2} a^{6} + \frac{1}{6} a^{5} - \frac{1}{3} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2}$, $\frac{1}{12} a^{12} - \frac{1}{6} a^{7} + \frac{1}{6} a^{6} - \frac{1}{2} a^{5} - \frac{1}{3} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4}$, $\frac{1}{12} a^{13} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{3} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4716} a^{14} - \frac{7}{4716} a^{13} + \frac{13}{786} a^{12} + \frac{4}{1179} a^{11} + \frac{1}{2358} a^{10} + \frac{15}{524} a^{9} - \frac{161}{4716} a^{8} + \frac{467}{1179} a^{7} + \frac{121}{262} a^{6} + \frac{3}{524} a^{5} + \frac{111}{524} a^{4} - \frac{157}{786} a^{3} + \frac{207}{524} a^{2} + \frac{61}{131} a - \frac{99}{524}$, $\frac{1}{1702476} a^{15} + \frac{173}{1702476} a^{14} - \frac{2533}{189164} a^{13} - \frac{58649}{1702476} a^{12} + \frac{221}{6498} a^{11} - \frac{5747}{283746} a^{10} + \frac{36715}{1702476} a^{9} + \frac{20635}{851238} a^{8} - \frac{10247}{141873} a^{7} - \frac{23202}{47291} a^{6} - \frac{278911}{567492} a^{5} - \frac{6481}{94582} a^{4} + \frac{76997}{189164} a^{3} - \frac{68999}{189164} a^{2} - \frac{2521}{94582} a + \frac{14799}{189164}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( \frac{101}{1572} a^{14} - \frac{707}{1572} a^{13} + \frac{3293}{1572} a^{12} - \frac{10567}{1572} a^{11} + \frac{29153}{1572} a^{10} - \frac{21917}{524} a^{9} + \frac{40779}{524} a^{8} - \frac{60731}{524} a^{7} + \frac{218275}{1572} a^{6} - \frac{207397}{1572} a^{5} + \frac{52497}{524} a^{4} - \frac{31059}{524} a^{3} + \frac{3923}{131} a^{2} - \frac{2989}{262} a + \frac{328}{131} \) (order $6$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 5400.30534286 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\wr C_2$ (as 16T39):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_2^2\wr C_2$
Character table for $C_2^2\wr C_2$

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{-7}) \), 4.0.1008.1, 4.0.3024.2 x2, 4.2.21168.1 x2, 4.0.1008.2, 4.0.432.1, 4.0.21168.1, \(\Q(\sqrt{-3}, \sqrt{-7})\), 8.0.448084224.8, 8.0.146313216.1, 8.0.146313216.2, 8.0.7169347584.8, 8.0.7169347584.4, 8.0.448084224.6, 8.0.49787136.3

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 R R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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$2.8.12.15$x^{8} + 2 x^{7} + 2 x^{4} + 12$$4$$2$$12$$C_2^2:C_4$$[2, 2]^{4}$
2.8.12.15$x^{8} + 2 x^{7} + 2 x^{4} + 12$$4$$2$$12$$C_2^2:C_4$$[2, 2]^{4}$
$3$3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
$7$7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$