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Properties

Label 16.8.13212017625...0000.1
Degree $16$
Signature $[8, 4]$
Discriminant $2^{24}\cdot 5^{8}\cdot 17^{10}$
Root discriminant $37.16$
Ramified primes $2, 5, 17$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times C_2^2.D_4$ (as 16T92)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-36, -456, -1870, -2584, 25, 3044, -292, -752, 1262, -304, -694, 104, 188, -8, -24, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 24*x^14 - 8*x^13 + 188*x^12 + 104*x^11 - 694*x^10 - 304*x^9 + 1262*x^8 - 752*x^7 - 292*x^6 + 3044*x^5 + 25*x^4 - 2584*x^3 - 1870*x^2 - 456*x - 36)
 
gp: K = bnfinit(x^16 - 24*x^14 - 8*x^13 + 188*x^12 + 104*x^11 - 694*x^10 - 304*x^9 + 1262*x^8 - 752*x^7 - 292*x^6 + 3044*x^5 + 25*x^4 - 2584*x^3 - 1870*x^2 - 456*x - 36, 1)
 

Normalized defining polynomial

\( x^{16} - 24 x^{14} - 8 x^{13} + 188 x^{12} + 104 x^{11} - 694 x^{10} - 304 x^{9} + 1262 x^{8} - 752 x^{7} - 292 x^{6} + 3044 x^{5} + 25 x^{4} - 2584 x^{3} - 1870 x^{2} - 456 x - 36 \)

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

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(13212017625982566400000000=2^{24}\cdot 5^{8}\cdot 17^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.16$
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:  $2, 5, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{4}$, $\frac{1}{6} a^{11} + \frac{1}{6} a^{9} + \frac{1}{6} a^{8} + \frac{1}{3} a^{7} - \frac{1}{2} a^{5} - \frac{1}{3} a^{4} - \frac{1}{2} a^{3} - \frac{1}{6} a^{2} + \frac{1}{3} a$, $\frac{1}{30} a^{12} + \frac{7}{30} a^{10} + \frac{7}{30} a^{9} - \frac{2}{15} a^{8} + \frac{1}{5} a^{7} + \frac{3}{10} a^{6} + \frac{2}{15} a^{5} - \frac{1}{10} a^{4} - \frac{7}{30} a^{3} + \frac{7}{15} a^{2} + \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{90} a^{13} + \frac{1}{45} a^{11} + \frac{7}{90} a^{10} + \frac{1}{15} a^{9} + \frac{1}{90} a^{8} - \frac{1}{90} a^{7} - \frac{13}{45} a^{6} - \frac{1}{5} a^{5} + \frac{11}{30} a^{4} - \frac{8}{45} a^{3} + \frac{41}{90} a^{2} - \frac{2}{45} a - \frac{1}{3}$, $\frac{1}{352440} a^{14} + \frac{229}{117480} a^{13} + \frac{2341}{176220} a^{12} + \frac{13741}{352440} a^{11} - \frac{1443}{7832} a^{10} + \frac{13177}{88110} a^{9} + \frac{4381}{32040} a^{8} + \frac{57037}{352440} a^{7} - \frac{128}{2937} a^{6} + \frac{45239}{117480} a^{5} + \frac{20035}{70488} a^{4} - \frac{30023}{176220} a^{3} + \frac{87079}{176220} a^{2} - \frac{3973}{9790} a + \frac{293}{1958}$, $\frac{1}{11417572580040} a^{15} + \frac{345853}{317154793890} a^{14} - \frac{863244109}{422873058520} a^{13} - \frac{186873594437}{11417572580040} a^{12} + \frac{2945960051}{518980571820} a^{11} + \frac{1247563557509}{11417572580040} a^{10} - \frac{2050785257}{128287332360} a^{9} - \frac{488529092221}{2854393145010} a^{8} + \frac{2687842469189}{11417572580040} a^{7} - \frac{3520519965671}{11417572580040} a^{6} + \frac{396082046581}{1427196572505} a^{5} + \frac{2557139219273}{11417572580040} a^{4} - \frac{142942487969}{2854393145010} a^{3} - \frac{73231947091}{518980571820} a^{2} - \frac{4694553278}{25949028591} a + \frac{21616747855}{190292876334}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $11$
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:  \( 14642543.1242 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_2^2.D_4$ (as 16T92):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 64
The 22 conjugacy class representatives for $C_2\times C_2^2.D_4$
Character table for $C_2\times C_2^2.D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{34}) \), \(\Q(\sqrt{85}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{170}) \), \(\Q(\sqrt{10}, \sqrt{17})\), \(\Q(\sqrt{2}, \sqrt{5})\), \(\Q(\sqrt{10}, \sqrt{34})\), \(\Q(\sqrt{2}, \sqrt{17})\), \(\Q(\sqrt{5}, \sqrt{17})\), \(\Q(\sqrt{2}, \sqrt{85})\), \(\Q(\sqrt{5}, \sqrt{34})\), 8.8.213813760000.1, 8.4.145393356800.2, 8.4.145393356800.4

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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$2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
2.4.6.1$x^{4} - 6 x^{2} + 4$$2$$2$$6$$C_2^2$$[3]^{2}$
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$17$17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$