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Properties

Label 16.0.25991671039...696.10
Degree $16$
Signature $[0, 8]$
Discriminant $2^{36}\cdot 3^{8}\cdot 7^{8}$
Root discriminant $21.80$
Ramified primes $2, 3, 7$
Class number $8$
Class group $[8]$
Galois group $C_2^2 \times D_4$ (as 16T25)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![169, -260, -250, -372, 1900, -352, -444, -4100, 6747, -3884, 912, -112, 112, -96, 38, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 38*x^14 - 96*x^13 + 112*x^12 - 112*x^11 + 912*x^10 - 3884*x^9 + 6747*x^8 - 4100*x^7 - 444*x^6 - 352*x^5 + 1900*x^4 - 372*x^3 - 250*x^2 - 260*x + 169)
 
gp: K = bnfinit(x^16 - 8*x^15 + 38*x^14 - 96*x^13 + 112*x^12 - 112*x^11 + 912*x^10 - 3884*x^9 + 6747*x^8 - 4100*x^7 - 444*x^6 - 352*x^5 + 1900*x^4 - 372*x^3 - 250*x^2 - 260*x + 169, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 38 x^{14} - 96 x^{13} + 112 x^{12} - 112 x^{11} + 912 x^{10} - 3884 x^{9} + 6747 x^{8} - 4100 x^{7} - 444 x^{6} - 352 x^{5} + 1900 x^{4} - 372 x^{3} - 250 x^{2} - 260 x + 169 \)

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:  \(2599167103947239325696=2^{36}\cdot 3^{8}\cdot 7^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.80$
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:  $2, 3, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{13} a^{11} - \frac{2}{13} a^{10} - \frac{2}{13} a^{9} - \frac{3}{13} a^{8} - \frac{6}{13} a^{7} + \frac{6}{13} a^{6} + \frac{6}{13} a^{5} - \frac{2}{13} a^{4} - \frac{6}{13} a^{3} + \frac{6}{13} a$, $\frac{1}{273} a^{12} - \frac{8}{273} a^{11} + \frac{12}{91} a^{10} + \frac{74}{273} a^{9} - \frac{22}{91} a^{8} - \frac{7}{39} a^{7} + \frac{5}{39} a^{6} + \frac{53}{273} a^{5} + \frac{58}{273} a^{4} + \frac{127}{273} a^{3} - \frac{1}{39} a^{2} - \frac{12}{91} a - \frac{1}{21}$, $\frac{1}{273} a^{13} - \frac{1}{39} a^{11} + \frac{47}{273} a^{10} - \frac{62}{273} a^{9} - \frac{94}{273} a^{8} + \frac{3}{13} a^{7} - \frac{29}{91} a^{6} + \frac{62}{273} a^{5} + \frac{1}{91} a^{4} + \frac{64}{273} a^{3} - \frac{92}{273} a^{2} + \frac{14}{39} a - \frac{8}{21}$, $\frac{1}{3549} a^{14} - \frac{4}{3549} a^{12} + \frac{2}{3549} a^{11} + \frac{1726}{3549} a^{10} - \frac{1741}{3549} a^{9} - \frac{206}{1183} a^{8} + \frac{419}{1183} a^{7} - \frac{1597}{3549} a^{6} + \frac{12}{1183} a^{5} - \frac{155}{507} a^{4} - \frac{1496}{3549} a^{3} + \frac{89}{507} a^{2} - \frac{110}{273} a + \frac{1}{7}$, $\frac{1}{20821494164289} a^{15} + \frac{437478941}{6940498054763} a^{14} - \frac{6006587402}{6940498054763} a^{13} - \frac{10195224398}{20821494164289} a^{12} - \frac{427621645852}{20821494164289} a^{11} - \frac{187765350239}{1601653397253} a^{10} - \frac{2059463772442}{20821494164289} a^{9} + \frac{357231868355}{20821494164289} a^{8} - \frac{1932728222029}{6940498054763} a^{7} + \frac{1528077098407}{20821494164289} a^{6} + \frac{350812512400}{2974499166327} a^{5} - \frac{1166112235952}{6940498054763} a^{4} - \frac{5764453704019}{20821494164289} a^{3} + \frac{793567995334}{6940498054763} a^{2} - \frac{603946146760}{1601653397253} a + \frac{48346377914}{123204107481}$

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

Class group and class number

$C_{8}$, which has order $8$

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:  \( -\frac{12928435599776}{2974499166327} a^{15} + \frac{214830345039675}{6940498054763} a^{14} - \frac{410445985986160}{2974499166327} a^{13} + \frac{2055311071139666}{6940498054763} a^{12} - \frac{4727490480817528}{20821494164289} a^{11} + \frac{461224112692790}{1601653397253} a^{10} - \frac{77268113984777698}{20821494164289} a^{9} + \frac{94545115353468761}{6940498054763} a^{8} - \frac{361691627538125138}{20821494164289} a^{7} + \frac{54058373141678924}{20821494164289} a^{6} + \frac{29079131143728944}{6940498054763} a^{5} + \frac{107921574382596974}{20821494164289} a^{4} - \frac{25705816375815034}{6940498054763} a^{3} - \frac{1598888334402803}{991499722109} a^{2} - \frac{172787345974618}{533884465751} a + \frac{103687600652858}{123204107481} \) (order $12$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 18213.1636607 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times D_4$ (as 16T25):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 20 conjugacy class representatives for $C_2^2 \times D_4$
Character table for $C_2^2 \times D_4$

Intermediate fields

\(\Q(\sqrt{42}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{14}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-14}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-42}) \), 4.0.4032.1, 4.0.1008.1, 4.0.4032.2, 4.0.1008.2, \(\Q(\sqrt{3}, \sqrt{-14})\), \(\Q(\sqrt{-3}, \sqrt{-14})\), \(\Q(i, \sqrt{14})\), \(\Q(\sqrt{3}, \sqrt{14})\), \(\Q(\zeta_{12})\), \(\Q(i, \sqrt{42})\), \(\Q(\sqrt{-3}, \sqrt{14})\), 8.0.12745506816.9, 8.0.12745506816.18, 8.0.12745506816.12, 8.0.260112384.8, 8.0.16257024.2, 8.0.12745506816.14, 8.0.12745506816.17

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

Sibling fields

Galois closure: data not computed
Degree 16 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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
2Data not computed
$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}$
3.4.2.1$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}$
$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}$
7.4.2.1$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}$