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Properties

Label 16.0.34552224922...5625.9
Degree $16$
Signature $[0, 8]$
Discriminant $5^{10}\cdot 29^{12}$
Root discriminant $34.17$
Ramified primes $5, 29$
Class number $12$ (GRH)
Class group $[12]$ (GRH)
Galois group $(C_2\times C_4).D_4$ (as 16T121)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![132119, 53788, -16340, 30159, 24954, -17395, 29363, -16545, 12495, -5307, 2866, -916, 386, -88, 29, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 29*x^14 - 88*x^13 + 386*x^12 - 916*x^11 + 2866*x^10 - 5307*x^9 + 12495*x^8 - 16545*x^7 + 29363*x^6 - 17395*x^5 + 24954*x^4 + 30159*x^3 - 16340*x^2 + 53788*x + 132119)
 
gp: K = bnfinit(x^16 - 4*x^15 + 29*x^14 - 88*x^13 + 386*x^12 - 916*x^11 + 2866*x^10 - 5307*x^9 + 12495*x^8 - 16545*x^7 + 29363*x^6 - 17395*x^5 + 24954*x^4 + 30159*x^3 - 16340*x^2 + 53788*x + 132119, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 29 x^{14} - 88 x^{13} + 386 x^{12} - 916 x^{11} + 2866 x^{10} - 5307 x^{9} + 12495 x^{8} - 16545 x^{7} + 29363 x^{6} - 17395 x^{5} + 24954 x^{4} + 30159 x^{3} - 16340 x^{2} + 53788 x + 132119 \)

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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{91} a^{14} - \frac{4}{13} a^{13} + \frac{40}{91} a^{12} - \frac{12}{91} a^{11} - \frac{1}{7} a^{10} - \frac{43}{91} a^{9} + \frac{24}{91} a^{8} - \frac{4}{13} a^{7} + \frac{33}{91} a^{6} - \frac{12}{91} a^{5} + \frac{12}{91} a^{4} - \frac{2}{13} a^{3} - \frac{23}{91} a^{2} + \frac{16}{91} a + \frac{2}{7}$, $\frac{1}{2512109953478184473267736025843973} a^{15} + \frac{256331515359886360776780346946}{193239227190629574866748925064921} a^{14} - \frac{1178486039782653467008329141801316}{2512109953478184473267736025843973} a^{13} + \frac{1073177927799414171046679578229658}{2512109953478184473267736025843973} a^{12} + \frac{85570991058108041555260480632542}{358872850496883496181105146549139} a^{11} - \frac{1173369313532784070892745775663253}{2512109953478184473267736025843973} a^{10} - \frac{4661646485377385346415387596347}{358872850496883496181105146549139} a^{9} - \frac{908294696783645777766398069312682}{2512109953478184473267736025843973} a^{8} - \frac{1209809985574344342879977770315036}{2512109953478184473267736025843973} a^{7} + \frac{824873809129106069359236640323428}{2512109953478184473267736025843973} a^{6} - \frac{1174174078513039066560799578601672}{2512109953478184473267736025843973} a^{5} + \frac{521066872952142087083811549536685}{2512109953478184473267736025843973} a^{4} - \frac{454495148994366240824539779325437}{2512109953478184473267736025843973} a^{3} + \frac{794637067985948926289991817011631}{2512109953478184473267736025843973} a^{2} + \frac{316035340669968374603716558201243}{2512109953478184473267736025843973} a - \frac{32747010367812571210269891667996}{193239227190629574866748925064921}$

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

Class group and class number

$C_{12}$, which has order $12$ (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:  \( 53193.9176201 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times C_4).D_4$ (as 16T121):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 28 conjugacy class representatives for $(C_2\times C_4).D_4$
Character table for $(C_2\times C_4).D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{29}) \), 4.4.4205.1, 4.0.609725.2, 4.0.121945.1, 8.4.88410125.1, 8.4.1858822878125.1, 8.0.371764575625.4

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$5$5.4.3.4$x^{4} + 40$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.4$x^{4} + 40$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
29Data not computed