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Properties

Label 16.8.97015201670...0625.1
Degree $16$
Signature $[8, 4]$
Discriminant $5^{8}\cdot 19^{6}\cdot 29^{6}\cdot 31^{6}$
Root discriminant $86.43$
Ramified primes $5, 19, 29, 31$
Class number $16$ (GRH)
Class group $[2, 2, 4]$ (GRH)
Galois group $C_2^3.(C_2\times D_4)$ (as 16T408)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![25701709, 2912506, -50712166, -47680261, -11552838, 6153536, 3248683, -331809, -189820, 61155, 3608, -6306, 258, 238, -39, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 39*x^14 + 238*x^13 + 258*x^12 - 6306*x^11 + 3608*x^10 + 61155*x^9 - 189820*x^8 - 331809*x^7 + 3248683*x^6 + 6153536*x^5 - 11552838*x^4 - 47680261*x^3 - 50712166*x^2 + 2912506*x + 25701709)
 
gp: K = bnfinit(x^16 - 4*x^15 - 39*x^14 + 238*x^13 + 258*x^12 - 6306*x^11 + 3608*x^10 + 61155*x^9 - 189820*x^8 - 331809*x^7 + 3248683*x^6 + 6153536*x^5 - 11552838*x^4 - 47680261*x^3 - 50712166*x^2 + 2912506*x + 25701709, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 39 x^{14} + 238 x^{13} + 258 x^{12} - 6306 x^{11} + 3608 x^{10} + 61155 x^{9} - 189820 x^{8} - 331809 x^{7} + 3248683 x^{6} + 6153536 x^{5} - 11552838 x^{4} - 47680261 x^{3} - 50712166 x^{2} + 2912506 x + 25701709 \)

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:  \(9701520167019972645874406640625=5^{8}\cdot 19^{6}\cdot 29^{6}\cdot 31^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $86.43$
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:  $5, 19, 29, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3509} a^{14} - \frac{17}{319} a^{13} + \frac{1602}{3509} a^{12} - \frac{845}{3509} a^{11} + \frac{7}{121} a^{10} + \frac{648}{3509} a^{9} + \frac{1546}{3509} a^{8} + \frac{1117}{3509} a^{7} - \frac{1717}{3509} a^{6} - \frac{74}{3509} a^{5} + \frac{1746}{3509} a^{4} - \frac{1212}{3509} a^{3} - \frac{762}{3509} a^{2} - \frac{700}{3509} a + \frac{133}{319}$, $\frac{1}{8528968467456676615188936611423783077728177300378616991} a^{15} - \frac{619756123539439415113388921715324971363752288959458}{8528968467456676615188936611423783077728177300378616991} a^{14} - \frac{940120492153407836800721411611001532092145730349192459}{8528968467456676615188936611423783077728177300378616991} a^{13} - \frac{3792652648915675598365641138191714500286407300734748217}{8528968467456676615188936611423783077728177300378616991} a^{12} - \frac{3583736349317036991080309340945403713210596682414340042}{8528968467456676615188936611423783077728177300378616991} a^{11} + \frac{3160244010859134207375586868274037077446634007160318350}{8528968467456676615188936611423783077728177300378616991} a^{10} + \frac{4109662148205871639535089582674069706336601411324173624}{8528968467456676615188936611423783077728177300378616991} a^{9} - \frac{77271909377209786565547607784975628196796469576322163}{775360769768788783198994237402162097975288845488965181} a^{8} - \frac{1199019414956425506328509390793373589368055227187397987}{8528968467456676615188936611423783077728177300378616991} a^{7} - \frac{2180145302956907349218103363508139208481560441652330226}{8528968467456676615188936611423783077728177300378616991} a^{6} - \frac{2856927773969241153503035924577689851893226806044955904}{8528968467456676615188936611423783077728177300378616991} a^{5} - \frac{46897266529772942108726679854551767544312146228131605}{95831106375917714777403782150829023345260419105377719} a^{4} - \frac{1720535492858433192306642647723109785439127817716936904}{8528968467456676615188936611423783077728177300378616991} a^{3} - \frac{3230804090129874567105389215347941260776303303485831070}{8528968467456676615188936611423783077728177300378616991} a^{2} + \frac{3379845683489649687424631394374223991631024619404182403}{8528968467456676615188936611423783077728177300378616991} a - \frac{63627402662965131445390186276108640059217695451652281}{775360769768788783198994237402162097975288845488965181}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{4}$, which has order $16$ (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:  \( 220644039.662 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.(C_2\times D_4)$ (as 16T408):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 26 conjugacy class representatives for $C_2^3.(C_2\times D_4)$
Character table for $C_2^3.(C_2\times D_4)$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.14725.1, 4.4.427025.2, 4.4.725.1, 8.4.107404356518125.2, 8.4.107404356518125.1, 8.8.182350350625.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ R R ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$19$19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.4.3.2$x^{4} - 19$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
19.4.3.2$x^{4} - 19$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
$29$29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
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}$
$31$31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.4.3.2$x^{4} - 31$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
31.4.3.2$x^{4} - 31$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$