↑

Properties

Label 16.0.10931326797...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{24}\cdot 3^{10}\cdot 5^{8}\cdot 7^{10}$
Root discriminant $42.40$
Ramified primes $2, 3, 5, 7$
Class number $72$ (GRH)
Class group $[2, 2, 18]$ (GRH)
Galois group $C_2\wr C_2^2$ (as 16T127)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![175876, -222632, 49752, 34840, 33676, -34112, 772, 3176, 2853, -1534, -280, 94, 109, -22, -8, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^14 - 22*x^13 + 109*x^12 + 94*x^11 - 280*x^10 - 1534*x^9 + 2853*x^8 + 3176*x^7 + 772*x^6 - 34112*x^5 + 33676*x^4 + 34840*x^3 + 49752*x^2 - 222632*x + 175876)
 
gp: K = bnfinit(x^16 - 8*x^14 - 22*x^13 + 109*x^12 + 94*x^11 - 280*x^10 - 1534*x^9 + 2853*x^8 + 3176*x^7 + 772*x^6 - 34112*x^5 + 33676*x^4 + 34840*x^3 + 49752*x^2 - 222632*x + 175876, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{14} - 22 x^{13} + 109 x^{12} + 94 x^{11} - 280 x^{10} - 1534 x^{9} + 2853 x^{8} + 3176 x^{7} + 772 x^{6} - 34112 x^{5} + 33676 x^{4} + 34840 x^{3} + 49752 x^{2} - 222632 x + 175876 \)

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:  \(109313267978738073600000000=2^{24}\cdot 3^{10}\cdot 5^{8}\cdot 7^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $42.40$
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:  $2, 3, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5}$, $\frac{1}{16} a^{12} + \frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{16} a^{8} - \frac{3}{8} a^{7} - \frac{3}{8} a^{6} - \frac{1}{8} a^{5} - \frac{5}{16} a^{4} - \frac{1}{8} a^{2} + \frac{1}{8}$, $\frac{1}{176} a^{13} + \frac{1}{88} a^{12} + \frac{13}{88} a^{11} + \frac{21}{88} a^{10} + \frac{19}{176} a^{9} - \frac{2}{11} a^{8} - \frac{37}{88} a^{7} + \frac{5}{88} a^{6} + \frac{5}{16} a^{5} + \frac{27}{88} a^{4} - \frac{9}{88} a^{3} + \frac{19}{44} a^{2} + \frac{25}{88} a + \frac{17}{44}$, $\frac{1}{12144} a^{14} - \frac{1}{12144} a^{13} - \frac{1}{6072} a^{12} + \frac{563}{3036} a^{11} + \frac{2401}{12144} a^{10} + \frac{1363}{12144} a^{9} + \frac{21}{92} a^{8} - \frac{701}{3036} a^{7} + \frac{1301}{12144} a^{6} + \frac{1517}{12144} a^{5} - \frac{305}{2024} a^{4} - \frac{1343}{6072} a^{3} + \frac{2221}{6072} a^{2} + \frac{45}{2024} a + \frac{233}{1518}$, $\frac{1}{2170200490312038378055270992} a^{15} - \frac{83047370494713492544373}{2170200490312038378055270992} a^{14} - \frac{5854941223572759401365765}{2170200490312038378055270992} a^{13} + \frac{6105245961678616584843223}{2170200490312038378055270992} a^{12} - \frac{28942778448355782018231541}{2170200490312038378055270992} a^{11} - \frac{29897139073752736859160973}{241133387812448708672807888} a^{10} + \frac{293226219414624144069101849}{2170200490312038378055270992} a^{9} + \frac{63616994346392486366962915}{2170200490312038378055270992} a^{8} + \frac{26703587222652425774960087}{197290953664730761641388272} a^{7} - \frac{81025015755897654047757299}{241133387812448708672807888} a^{6} + \frac{1003266063473610929902499611}{2170200490312038378055270992} a^{5} + \frac{112582253644036917500301299}{2170200490312038378055270992} a^{4} + \frac{5372927090208977835208732}{15070836738278044292050493} a^{3} - \frac{31805001489488848876302500}{135637530644502398628454437} a^{2} - \frac{239575982403422739127712407}{1085100245156019189027635496} a + \frac{16414603066604767622056589}{47178271528522573435984152}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{18}$, which has order $72$ (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:  \( 63425.6271299 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\wr C_2^2$ (as 16T127):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 16 conjugacy class representatives for $C_2\wr C_2^2$
Character table for $C_2\wr C_2^2$

Intermediate fields

\(\Q(\sqrt{35}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{105}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{3}, \sqrt{5})\), \(\Q(\sqrt{15}, \sqrt{21})\), \(\Q(\sqrt{7}, \sqrt{15})\), \(\Q(\sqrt{3}, \sqrt{35})\), \(\Q(\sqrt{3}, \sqrt{7})\), \(\Q(\sqrt{5}, \sqrt{7})\), \(\Q(\sqrt{5}, \sqrt{21})\), 8.0.10455298560000.17, 8.0.10455298560000.20, 8.8.31116960000.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 8 siblings: data not computed
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 R ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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
$2$2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.8.16.4$x^{8} + 6 x^{6} + 6 x^{4} + 8 x^{3} + 4 x^{2} + 8 x + 20$$4$$2$$16$$D_4$$[2, 3]^{2}$
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
$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}$
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.8.6.2$x^{8} - 49 x^{4} + 3969$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$