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Properties

Label 16.0.10271172687...5625.4
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 29^{10}$
Root discriminant $27.43$
Ramified primes $5, 29$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $C_2^2.D_4$ (as 16T33)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![400, -3400, 18760, -35200, 27621, -7747, 1698, -4526, 3108, -199, -143, -226, 113, 11, -7, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - 7*x^14 + 11*x^13 + 113*x^12 - 226*x^11 - 143*x^10 - 199*x^9 + 3108*x^8 - 4526*x^7 + 1698*x^6 - 7747*x^5 + 27621*x^4 - 35200*x^3 + 18760*x^2 - 3400*x + 400)
 
gp: K = bnfinit(x^16 - 3*x^15 - 7*x^14 + 11*x^13 + 113*x^12 - 226*x^11 - 143*x^10 - 199*x^9 + 3108*x^8 - 4526*x^7 + 1698*x^6 - 7747*x^5 + 27621*x^4 - 35200*x^3 + 18760*x^2 - 3400*x + 400, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} - 7 x^{14} + 11 x^{13} + 113 x^{12} - 226 x^{11} - 143 x^{10} - 199 x^{9} + 3108 x^{8} - 4526 x^{7} + 1698 x^{6} - 7747 x^{5} + 27621 x^{4} - 35200 x^{3} + 18760 x^{2} - 3400 x + 400 \)

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:  \(102711726879931884765625=5^{12}\cdot 29^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.43$
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:  $5, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{8} - \frac{1}{3}$, $\frac{1}{6} a^{9} - \frac{1}{6} a^{8} - \frac{1}{6} a^{7} - \frac{1}{3} a^{6} - \frac{1}{6} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{30} a^{10} + \frac{2}{15} a^{8} - \frac{1}{6} a^{7} + \frac{11}{30} a^{6} - \frac{1}{6} a^{5} + \frac{7}{15} a^{4} + \frac{1}{3} a^{3} + \frac{1}{30} a^{2} - \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{30} a^{11} - \frac{1}{30} a^{9} - \frac{2}{15} a^{7} - \frac{1}{6} a^{6} - \frac{1}{30} a^{5} + \frac{1}{3} a^{4} + \frac{1}{30} a^{3} - \frac{1}{6} a^{2} - \frac{1}{2} a + \frac{1}{3}$, $\frac{1}{90} a^{12} - \frac{1}{90} a^{11} + \frac{1}{90} a^{9} + \frac{2}{45} a^{7} + \frac{7}{18} a^{6} + \frac{8}{45} a^{5} - \frac{7}{18} a^{4} + \frac{7}{45} a^{3} + \frac{11}{90} a^{2} - \frac{1}{9}$, $\frac{1}{90} a^{13} - \frac{1}{90} a^{11} + \frac{1}{90} a^{10} + \frac{1}{90} a^{9} + \frac{2}{45} a^{8} + \frac{1}{10} a^{7} - \frac{1}{10} a^{6} + \frac{41}{90} a^{5} + \frac{1}{10} a^{4} - \frac{1}{18} a^{3} + \frac{41}{90} a^{2} - \frac{4}{9} a + \frac{2}{9}$, $\frac{1}{180} a^{14} - \frac{1}{180} a^{13} - \frac{1}{180} a^{12} - \frac{1}{180} a^{11} - \frac{1}{60} a^{10} + \frac{1}{30} a^{9} - \frac{7}{180} a^{8} + \frac{1}{20} a^{7} - \frac{29}{90} a^{6} - \frac{7}{90} a^{5} - \frac{43}{90} a^{4} - \frac{77}{180} a^{3} - \frac{23}{60} a^{2} + \frac{1}{6} a - \frac{4}{9}$, $\frac{1}{167839960433400} a^{15} + \frac{263822551699}{167839960433400} a^{14} + \frac{52656420391}{167839960433400} a^{13} + \frac{261352579991}{55946653477800} a^{12} + \frac{445989588953}{55946653477800} a^{11} - \frac{779440709}{975813723450} a^{10} - \frac{256033405871}{18648884492600} a^{9} - \frac{20081367210637}{167839960433400} a^{8} - \frac{243085918631}{1951627446900} a^{7} - \frac{9011507712599}{83919980216700} a^{6} + \frac{24708807781351}{83919980216700} a^{5} - \frac{25728238908001}{55946653477800} a^{4} + \frac{745409240827}{3729776898520} a^{3} - \frac{3555183672493}{16783996043340} a^{2} + \frac{87071196877}{279733267389} a + \frac{76395471272}{839199802167}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  \( 73371.5821514 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2.D_4$ (as 16T33):

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 $C_2^2.D_4$
Character table for $C_2^2.D_4$

Intermediate fields

\(\Q(\sqrt{145}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{29}) \), 4.0.105125.1 x2, 4.0.3625.1 x2, \(\Q(\sqrt{5}, \sqrt{29})\), 8.4.64097340625.2 x2, 8.0.11051265625.4, 8.0.11051265625.2 x2

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
$29$29.4.3.2$x^{4} - 116$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.3.2$x^{4} - 116$$4$$1$$3$$C_4$$[\ ]_{4}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$