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Properties

Label 16.6.20423375579...9375.1
Degree $16$
Signature $[6, 5]$
Discriminant $-\,5^{12}\cdot 101^{6}\cdot 199^{3}$
Root discriminant $50.92$
Ramified primes $5, 101, 199$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 16T1643

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![22081, 120130, 184269, 83485, 88497, 226775, 230492, 125525, 47045, 13050, 1673, -410, -203, -70, -19, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 19*x^14 - 70*x^13 - 203*x^12 - 410*x^11 + 1673*x^10 + 13050*x^9 + 47045*x^8 + 125525*x^7 + 230492*x^6 + 226775*x^5 + 88497*x^4 + 83485*x^3 + 184269*x^2 + 120130*x + 22081)
 
gp: K = bnfinit(x^16 - 19*x^14 - 70*x^13 - 203*x^12 - 410*x^11 + 1673*x^10 + 13050*x^9 + 47045*x^8 + 125525*x^7 + 230492*x^6 + 226775*x^5 + 88497*x^4 + 83485*x^3 + 184269*x^2 + 120130*x + 22081, 1)
 

Normalized defining polynomial

\( x^{16} - 19 x^{14} - 70 x^{13} - 203 x^{12} - 410 x^{11} + 1673 x^{10} + 13050 x^{9} + 47045 x^{8} + 125525 x^{7} + 230492 x^{6} + 226775 x^{5} + 88497 x^{4} + 83485 x^{3} + 184269 x^{2} + 120130 x + 22081 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-2042337557936057128662109375=-\,5^{12}\cdot 101^{6}\cdot 199^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $50.92$
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:  $5, 101, 199$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} - \frac{2}{5} a^{6} - \frac{1}{5} a^{4} + \frac{2}{5} a^{2} + \frac{1}{5}$, $\frac{1}{5} a^{9} - \frac{2}{5} a^{7} - \frac{1}{5} a^{5} + \frac{2}{5} a^{3} + \frac{1}{5} a$, $\frac{1}{5} a^{10} + \frac{2}{5}$, $\frac{1}{5} a^{11} + \frac{2}{5} a$, $\frac{1}{5} a^{12} + \frac{2}{5} a^{2}$, $\frac{1}{5} a^{13} + \frac{2}{5} a^{3}$, $\frac{1}{5} a^{14} + \frac{2}{5} a^{4}$, $\frac{1}{2109145757923704773391447297153418145} a^{15} + \frac{74305612081895281499053330843106816}{2109145757923704773391447297153418145} a^{14} - \frac{111444628509273505320607367330536626}{2109145757923704773391447297153418145} a^{13} - \frac{9468086361268372786848298072722159}{2109145757923704773391447297153418145} a^{12} - \frac{9763359662580258938093119077809937}{191740523447609524853767936104856195} a^{11} + \frac{21098128778799695360487641540524961}{2109145757923704773391447297153418145} a^{10} + \frac{161508517965130345656827246201878176}{2109145757923704773391447297153418145} a^{9} - \frac{7877827631465837242887653462768282}{421829151584740954678289459430683629} a^{8} - \frac{946327585089819022032443831543185727}{2109145757923704773391447297153418145} a^{7} + \frac{5455491033611377200595039125971984}{421829151584740954678289459430683629} a^{6} + \frac{73832497136184633682838282306072366}{2109145757923704773391447297153418145} a^{5} + \frac{112277958919480797518143348568125282}{2109145757923704773391447297153418145} a^{4} - \frac{205927641435087803950452566371609992}{421829151584740954678289459430683629} a^{3} + \frac{24195980565000160093300142648733312}{191740523447609524853767936104856195} a^{2} + \frac{909931620838805224469424797133418442}{2109145757923704773391447297153418145} a + \frac{600403673245641433874194775682884557}{2109145757923704773391447297153418145}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $10$
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:  \( 14058426.6164 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

16T1643:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4096
The 64 conjugacy class representatives for t16n1643 are not computed
Character table for t16n1643 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.2525.1, 8.8.16098453125.1

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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ $16$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$101$101.2.1.2$x^{2} + 202$$2$$1$$1$$C_2$$[\ ]_{2}$
101.2.1.2$x^{2} + 202$$2$$1$$1$$C_2$$[\ ]_{2}$
101.4.0.1$x^{4} - x + 12$$1$$4$$0$$C_4$$[\ ]^{4}$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
101.4.2.1$x^{4} + 505 x^{2} + 91809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$199$$\Q_{199}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{199}$$x + 2$$1$$1$$0$Trivial$[\ ]$
199.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
199.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
199.2.1.2$x^{2} + 398$$2$$1$$1$$C_2$$[\ ]_{2}$
199.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
199.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
199.4.2.1$x^{4} + 2189 x^{2} + 1425636$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$