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Properties

Label 16.0.88041439641...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 3^{8}\cdot 5^{12}\cdot 181^{2}$
Root discriminant $31.37$
Ramified primes $2, 3, 5, 181$
Class number $48$ (GRH)
Class group $[2, 24]$ (GRH)
Galois group $C_2^4.C_2^3$ (as 16T208)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![10141, -43324, 120792, -152794, 139463, -104914, 64076, -35362, 17493, -7514, 3164, -1030, 373, -88, 28, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 + 28*x^14 - 88*x^13 + 373*x^12 - 1030*x^11 + 3164*x^10 - 7514*x^9 + 17493*x^8 - 35362*x^7 + 64076*x^6 - 104914*x^5 + 139463*x^4 - 152794*x^3 + 120792*x^2 - 43324*x + 10141)
 
gp: K = bnfinit(x^16 - 4*x^15 + 28*x^14 - 88*x^13 + 373*x^12 - 1030*x^11 + 3164*x^10 - 7514*x^9 + 17493*x^8 - 35362*x^7 + 64076*x^6 - 104914*x^5 + 139463*x^4 - 152794*x^3 + 120792*x^2 - 43324*x + 10141, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} + 28 x^{14} - 88 x^{13} + 373 x^{12} - 1030 x^{11} + 3164 x^{10} - 7514 x^{9} + 17493 x^{8} - 35362 x^{7} + 64076 x^{6} - 104914 x^{5} + 139463 x^{4} - 152794 x^{3} + 120792 x^{2} - 43324 x + 10141 \)

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:  \(880414396416000000000000=2^{24}\cdot 3^{8}\cdot 5^{12}\cdot 181^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.37$
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:  $2, 3, 5, 181$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{179} a^{14} + \frac{71}{179} a^{13} + \frac{53}{179} a^{12} + \frac{86}{179} a^{11} - \frac{28}{179} a^{10} + \frac{26}{179} a^{9} - \frac{68}{179} a^{8} - \frac{54}{179} a^{7} - \frac{88}{179} a^{6} + \frac{82}{179} a^{5} - \frac{16}{179} a^{4} + \frac{45}{179} a^{3} - \frac{50}{179} a^{2} + \frac{9}{179} a + \frac{6}{179}$, $\frac{1}{6177950470012261388498023439} a^{15} - \frac{10955281769286209631828150}{6177950470012261388498023439} a^{14} + \frac{1333578688787386267890068803}{6177950470012261388498023439} a^{13} - \frac{398667902538780797943130363}{6177950470012261388498023439} a^{12} - \frac{1227183101463618487304763363}{6177950470012261388498023439} a^{11} + \frac{746120514678980742660876502}{6177950470012261388498023439} a^{10} - \frac{1069647082576698700630617426}{6177950470012261388498023439} a^{9} + \frac{2505745712556008589962872732}{6177950470012261388498023439} a^{8} + \frac{2809252139206926052848483860}{6177950470012261388498023439} a^{7} + \frac{1301284346163722545745720254}{6177950470012261388498023439} a^{6} + \frac{1540598872110538614398379192}{6177950470012261388498023439} a^{5} + \frac{770029284701821794639182554}{6177950470012261388498023439} a^{4} - \frac{2920763494596236996315333414}{6177950470012261388498023439} a^{3} - \frac{1940994948831461478909556842}{6177950470012261388498023439} a^{2} - \frac{973540082068105698401144355}{6177950470012261388498023439} a + \frac{1048863136705104334593312073}{6177950470012261388498023439}$

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

Class group and class number

$C_{2}\times C_{24}$, which has order $48$ (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:  \( 3121.7160225 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4.C_2^3$ (as 16T208):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 128
The 32 conjugacy class representatives for $C_2^4.C_2^3$
Character table for $C_2^4.C_2^3$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{15}) \), \(\Q(\zeta_{20})^+\), \(\Q(\zeta_{15})^+\), \(\Q(\sqrt{3}, \sqrt{5})\), 8.0.37532160000.1, 8.0.938304000000.1, \(\Q(\zeta_{60})^+\)

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 R R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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
2Data not computed
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$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}$
$181$$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
181.2.1.1$x^{2} - 181$$2$$1$$1$$C_2$$[\ ]_{2}$
181.2.1.1$x^{2} - 181$$2$$1$$1$$C_2$$[\ ]_{2}$