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Properties

Label 16.0.46813515740...0000.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{38}\cdot 5^{12}\cdot 17^{8}$
Root discriminant $71.52$
Ramified primes $2, 5, 17$
Class number $8$ (GRH)
Class group $[2, 4]$ (GRH)
Galois group $(C_2\times D_4):C_4$ (as 16T120)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9235031, -10705860, -2476140, 5803680, 524223, -1774300, -67990, 298520, 27644, -34100, -5340, 2320, 617, -100, -30, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 30*x^14 - 100*x^13 + 617*x^12 + 2320*x^11 - 5340*x^10 - 34100*x^9 + 27644*x^8 + 298520*x^7 - 67990*x^6 - 1774300*x^5 + 524223*x^4 + 5803680*x^3 - 2476140*x^2 - 10705860*x + 9235031)
 
gp: K = bnfinit(x^16 - 30*x^14 - 100*x^13 + 617*x^12 + 2320*x^11 - 5340*x^10 - 34100*x^9 + 27644*x^8 + 298520*x^7 - 67990*x^6 - 1774300*x^5 + 524223*x^4 + 5803680*x^3 - 2476140*x^2 - 10705860*x + 9235031, 1)
 

Normalized defining polynomial

\( x^{16} - 30 x^{14} - 100 x^{13} + 617 x^{12} + 2320 x^{11} - 5340 x^{10} - 34100 x^{9} + 27644 x^{8} + 298520 x^{7} - 67990 x^{6} - 1774300 x^{5} + 524223 x^{4} + 5803680 x^{3} - 2476140 x^{2} - 10705860 x + 9235031 \)

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:  \(468135157405057024000000000000=2^{38}\cdot 5^{12}\cdot 17^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $71.52$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{6} a^{12} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{2} + \frac{1}{6}$, $\frac{1}{6} a^{13} + \frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{6} a$, $\frac{1}{12} a^{14} - \frac{1}{12} a^{12} - \frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{6} a^{4} + \frac{1}{12} a^{2} - \frac{1}{4}$, $\frac{1}{1580446873263647389827869351352696} a^{15} + \frac{15085764131210628333863029248699}{526815624421215796609289783784232} a^{14} + \frac{31873882167861892657729996574093}{526815624421215796609289783784232} a^{13} - \frac{18893787007511706287746342836049}{1580446873263647389827869351352696} a^{12} + \frac{24090868022275381522844217075025}{65851953052651974576161222973029} a^{11} + \frac{98205884510484362951764852982993}{197555859157955923728483668919087} a^{10} + \frac{83854776165430433897918732321429}{395111718315911847456967337838174} a^{9} + \frac{12226674732895122026371142672929}{65851953052651974576161222973029} a^{8} + \frac{12929615322589813270358011866733}{131703906105303949152322445946058} a^{7} - \frac{144493606203160599560045193311933}{395111718315911847456967337838174} a^{6} - \frac{86576332520879466011695214369539}{263407812210607898304644891892116} a^{5} + \frac{126020605075120993367888019805075}{263407812210607898304644891892116} a^{4} - \frac{720176518878632912274346039279231}{1580446873263647389827869351352696} a^{3} + \frac{50706506095015527542695313528787}{526815624421215796609289783784232} a^{2} - \frac{313949440657629336327845731085519}{1580446873263647389827869351352696} a + \frac{8819483289607316192303896173}{573248775213510116005756021528}$

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

Class group and class number

$C_{2}\times C_{4}$, 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:  \( -1 \) (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:  \( 85271111.0063 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$(C_2\times D_4):C_4$ (as 16T120):

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 $(C_2\times D_4):C_4$
Character table for $(C_2\times D_4):C_4$ is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{170}) \), 4.0.2312.1, 4.0.57800.1, \(\Q(\sqrt{10}, \sqrt{17})\), 8.0.213813760000.23

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$2.4.11.10$x^{4} + 8 x + 6$$4$$1$$11$$C_4$$[3, 4]$
2.4.11.8$x^{4} + 4 x^{2} + 10$$4$$1$$11$$C_4$$[3, 4]$
2.8.16.3$x^{8} + 2 x^{6} + 6 x^{4} + 4 x^{2} + 8 x + 28$$4$$2$$16$$C_4\times C_2$$[2, 3]^{2}$
5Data not computed
$17$17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$