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Properties

Label 16.0.34552224922...5625.8
Degree $16$
Signature $[0, 8]$
Discriminant $5^{10}\cdot 29^{12}$
Root discriminant $34.17$
Ramified primes $5, 29$
Class number $36$ (GRH)
Class group $[3, 12]$ (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![7295, -13835, 11971, -7738, 6817, -6078, 3906, -1065, -90, 95, -70, 100, -19, 12, -3, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 3*x^15 - 3*x^14 + 12*x^13 - 19*x^12 + 100*x^11 - 70*x^10 + 95*x^9 - 90*x^8 - 1065*x^7 + 3906*x^6 - 6078*x^5 + 6817*x^4 - 7738*x^3 + 11971*x^2 - 13835*x + 7295)
 
gp: K = bnfinit(x^16 - 3*x^15 - 3*x^14 + 12*x^13 - 19*x^12 + 100*x^11 - 70*x^10 + 95*x^9 - 90*x^8 - 1065*x^7 + 3906*x^6 - 6078*x^5 + 6817*x^4 - 7738*x^3 + 11971*x^2 - 13835*x + 7295, 1)
 

Normalized defining polynomial

\( x^{16} - 3 x^{15} - 3 x^{14} + 12 x^{13} - 19 x^{12} + 100 x^{11} - 70 x^{10} + 95 x^{9} - 90 x^{8} - 1065 x^{7} + 3906 x^{6} - 6078 x^{5} + 6817 x^{4} - 7738 x^{3} + 11971 x^{2} - 13835 x + 7295 \)

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:  \(3455222492240908603515625=5^{10}\cdot 29^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{15} a^{10} + \frac{1}{15} a^{9} - \frac{2}{5} a^{7} + \frac{2}{5} a^{6} + \frac{2}{5} a^{4} - \frac{1}{5} a^{3} - \frac{1}{15} a^{2} + \frac{1}{3} a$, $\frac{1}{90} a^{11} - \frac{1}{30} a^{10} - \frac{7}{45} a^{9} + \frac{7}{45} a^{8} + \frac{1}{3} a^{7} - \frac{1}{10} a^{6} + \frac{7}{30} a^{5} - \frac{7}{15} a^{4} + \frac{13}{45} a^{3} - \frac{2}{5} a^{2} + \frac{7}{18} a - \frac{1}{18}$, $\frac{1}{270} a^{12} - \frac{1}{270} a^{11} - \frac{1}{135} a^{10} - \frac{13}{135} a^{9} - \frac{16}{135} a^{8} + \frac{11}{90} a^{7} - \frac{23}{90} a^{6} + \frac{1}{3} a^{5} + \frac{5}{27} a^{4} - \frac{19}{135} a^{3} + \frac{7}{54} a^{2} + \frac{19}{54} a + \frac{8}{27}$, $\frac{1}{810} a^{13} - \frac{1}{270} a^{11} - \frac{1}{81} a^{10} - \frac{4}{81} a^{9} + \frac{1}{810} a^{8} + \frac{7}{45} a^{7} + \frac{133}{270} a^{6} - \frac{13}{81} a^{5} - \frac{5}{27} a^{4} - \frac{109}{270} a^{3} + \frac{56}{405} a^{2} - \frac{55}{162} a - \frac{19}{81}$, $\frac{1}{36231300} a^{14} + \frac{421}{36231300} a^{13} + \frac{2917}{6038550} a^{12} + \frac{3683}{5175900} a^{11} - \frac{155333}{36231300} a^{10} - \frac{151253}{1341900} a^{9} + \frac{4341553}{36231300} a^{8} - \frac{104687}{3019275} a^{7} - \frac{12706993}{36231300} a^{6} + \frac{7116239}{36231300} a^{5} - \frac{1480939}{4025700} a^{4} - \frac{10042343}{36231300} a^{3} - \frac{5364983}{18115650} a^{2} - \frac{4339}{115020} a + \frac{2087761}{7246260}$, $\frac{1}{18559048649400} a^{15} + \frac{27613}{9279524324700} a^{14} - \frac{82056103}{261395051400} a^{13} - \frac{29330143909}{18559048649400} a^{12} + \frac{733165981}{257764564575} a^{11} - \frac{295738418}{39320018325} a^{10} - \frac{203432990143}{1325646332100} a^{9} - \frac{77483822099}{18559048649400} a^{8} + \frac{4678994671067}{18559048649400} a^{7} + \frac{2895889593797}{9279524324700} a^{6} + \frac{187156274968}{2319881081175} a^{5} - \frac{1247677378249}{9279524324700} a^{4} - \frac{216431730283}{687372172200} a^{3} + \frac{262324849369}{3711809729880} a^{2} - \frac{56060675653}{463976216235} a - \frac{268168794079}{742361945976}$

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

Class group and class number

$C_{3}\times C_{12}$, which has order $36$ (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

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{145}) \), \(\Q(\sqrt{29}) \), 4.0.121945.1 x2, \(\Q(\sqrt{5}, \sqrt{29})\), 4.0.609725.1 x2, 8.0.371764575625.3 x2, 8.4.64097340625.2 x2, 8.0.371764575625.5

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
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$29.8.6.1$x^{8} - 203 x^{4} + 68121$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
29.8.6.1$x^{8} - 203 x^{4} + 68121$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$