↑

Properties

Label 16.0.17958563260...0625.2
Degree $16$
Signature $[0, 8]$
Discriminant $3^{12}\cdot 5^{12}\cdot 7^{12}$
Root discriminant $32.80$
Ramified primes $3, 5, 7$
Class number $8$ (GRH)
Class group $[2, 2, 2]$ (GRH)
Galois group $C_4:C_4$ (as 16T8)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 5, 21, 30, 50, -75, 674, -700, 1239, 700, 674, 75, 50, -30, 21, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 5*x^15 + 21*x^14 - 30*x^13 + 50*x^12 + 75*x^11 + 674*x^10 + 700*x^9 + 1239*x^8 - 700*x^7 + 674*x^6 - 75*x^5 + 50*x^4 + 30*x^3 + 21*x^2 + 5*x + 1)
 
gp: K = bnfinit(x^16 - 5*x^15 + 21*x^14 - 30*x^13 + 50*x^12 + 75*x^11 + 674*x^10 + 700*x^9 + 1239*x^8 - 700*x^7 + 674*x^6 - 75*x^5 + 50*x^4 + 30*x^3 + 21*x^2 + 5*x + 1, 1)
 

Normalized defining polynomial

\( x^{16} - 5 x^{15} + 21 x^{14} - 30 x^{13} + 50 x^{12} + 75 x^{11} + 674 x^{10} + 700 x^{9} + 1239 x^{8} - 700 x^{7} + 674 x^{6} - 75 x^{5} + 50 x^{4} + 30 x^{3} + 21 x^{2} + 5 x + 1 \)

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:  \(1795856326022129150390625=3^{12}\cdot 5^{12}\cdot 7^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.80$
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:  $3, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{118} a^{12} - \frac{18}{59} a^{11} + \frac{7}{59} a^{10} - \frac{7}{118} a^{9} - \frac{26}{59} a^{8} - \frac{6}{59} a^{7} - \frac{25}{118} a^{6} + \frac{6}{59} a^{5} - \frac{26}{59} a^{4} + \frac{7}{118} a^{3} + \frac{7}{59} a^{2} + \frac{18}{59} a + \frac{1}{118}$, $\frac{1}{118} a^{13} + \frac{8}{59} a^{11} + \frac{25}{118} a^{10} + \frac{25}{59} a^{9} + \frac{2}{59} a^{8} + \frac{15}{118} a^{7} + \frac{28}{59} a^{6} + \frac{13}{59} a^{5} + \frac{23}{118} a^{4} + \frac{15}{59} a^{3} - \frac{25}{59} a^{2} - \frac{1}{118} a + \frac{18}{59}$, $\frac{1}{48660958} a^{14} + \frac{2235}{48660958} a^{13} + \frac{28925}{24330479} a^{12} + \frac{14119525}{48660958} a^{11} + \frac{13429297}{48660958} a^{10} + \frac{2564036}{24330479} a^{9} + \frac{4778495}{48660958} a^{8} + \frac{21279195}{48660958} a^{7} - \frac{7131629}{24330479} a^{6} + \frac{13788073}{48660958} a^{5} + \frac{17911659}{48660958} a^{4} + \frac{1492619}{24330479} a^{3} + \frac{11901199}{48660958} a^{2} + \frac{18971761}{48660958} a + \frac{11752858}{24330479}$, $\frac{1}{58441810558} a^{15} - \frac{27}{29220905279} a^{14} - \frac{25677115}{58441810558} a^{13} - \frac{85335756}{29220905279} a^{12} + \frac{13212981677}{29220905279} a^{11} - \frac{17530917903}{58441810558} a^{10} - \frac{2743934034}{29220905279} a^{9} + \frac{1944715782}{29220905279} a^{8} - \frac{2126586661}{58441810558} a^{7} + \frac{4656892898}{29220905279} a^{6} - \frac{490303383}{29220905279} a^{5} - \frac{6972070665}{58441810558} a^{4} + \frac{13201083341}{29220905279} a^{3} - \frac{1248666376}{29220905279} a^{2} - \frac{3662523005}{58441810558} a + \frac{728267135}{58441810558}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}$, which has order $8$ (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:  \( -\frac{475552}{10699709} a^{15} + \frac{7046945712}{29220905279} a^{14} - \frac{29813472528}{29220905279} a^{13} + \frac{49267848889}{29220905279} a^{12} - \frac{75808903632}{29220905279} a^{11} - \frac{80693067840}{29220905279} a^{10} - \frac{815626832048}{29220905279} a^{9} - \frac{530576321520}{29220905279} a^{8} - \frac{1079591379576}{29220905279} a^{7} + \frac{1635447771664}{29220905279} a^{6} - \frac{1067654960400}{29220905279} a^{5} + \frac{116001096288}{29220905279} a^{4} + \frac{217724831632}{29220905279} a^{3} - \frac{259621433400}{29220905279} a^{2} + \frac{33191877264}{29220905279} a + \frac{5496253264}{29220905279} \) (order $10$)
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:  \( 120012.452011 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4:C_4$ (as 16T8):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 16
The 10 conjugacy class representatives for $C_4:C_4$
Character table for $C_4:C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{105}) \), \(\Q(\sqrt{5}, \sqrt{21})\), 4.4.46305.1 x2, 4.4.231525.1 x2, 4.0.55125.1, \(\Q(\zeta_{5})\), 8.8.53603825625.1, 8.0.3038765625.3, 8.0.1340095640625.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ R R R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
3Data not computed
$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}$
7Data not computed