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Properties

Label 16.0.82432069367...0721.4
Degree $16$
Signature $[0, 8]$
Discriminant $13^{12}\cdot 29^{12}$
Root discriminant $85.56$
Ramified primes $13, 29$
Class number $360$ (GRH)
Class group $[2, 6, 30]$ (GRH)
Galois group $C_4:C_4$ (as 16T8)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4004113, -3357557, 12664385, 3617292, -3004289, -9376, 87135, -155387, 59327, -2149, -940, -162, 581, -66, -31, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 31*x^14 - 66*x^13 + 581*x^12 - 162*x^11 - 940*x^10 - 2149*x^9 + 59327*x^8 - 155387*x^7 + 87135*x^6 - 9376*x^5 - 3004289*x^4 + 3617292*x^3 + 12664385*x^2 - 3357557*x + 4004113)
 
gp: K = bnfinit(x^16 - 31*x^14 - 66*x^13 + 581*x^12 - 162*x^11 - 940*x^10 - 2149*x^9 + 59327*x^8 - 155387*x^7 + 87135*x^6 - 9376*x^5 - 3004289*x^4 + 3617292*x^3 + 12664385*x^2 - 3357557*x + 4004113, 1)
 

Normalized defining polynomial

\( x^{16} - 31 x^{14} - 66 x^{13} + 581 x^{12} - 162 x^{11} - 940 x^{10} - 2149 x^{9} + 59327 x^{8} - 155387 x^{7} + 87135 x^{6} - 9376 x^{5} - 3004289 x^{4} + 3617292 x^{3} + 12664385 x^{2} - 3357557 x + 4004113 \)

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:  \(8243206936713178643875538610721=13^{12}\cdot 29^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $85.56$
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:  $13, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{1624} a^{12} - \frac{19}{1624} a^{11} - \frac{93}{812} a^{10} + \frac{12}{203} a^{9} - \frac{41}{812} a^{8} - \frac{615}{1624} a^{7} - \frac{729}{1624} a^{6} + \frac{337}{812} a^{5} - \frac{109}{1624} a^{4} - \frac{195}{1624} a^{3} + \frac{307}{1624} a^{2} + \frac{741}{1624} a + \frac{663}{1624}$, $\frac{1}{1624} a^{13} + \frac{265}{1624} a^{11} - \frac{95}{812} a^{10} + \frac{59}{812} a^{9} + \frac{263}{1624} a^{8} - \frac{117}{812} a^{7} - \frac{185}{1624} a^{6} + \frac{517}{1624} a^{5} - \frac{321}{812} a^{4} + \frac{331}{812} a^{3} - \frac{367}{812} a^{2} - \frac{49}{116} a + \frac{417}{1624}$, $\frac{1}{1624} a^{14} - \frac{27}{1624} a^{11} - \frac{31}{406} a^{10} - \frac{5}{1624} a^{9} + \frac{48}{203} a^{8} + \frac{195}{812} a^{7} - \frac{183}{812} a^{6} + \frac{25}{203} a^{5} - \frac{71}{232} a^{4} + \frac{597}{1624} a^{3} + \frac{27}{56} a^{2} - \frac{32}{203} a - \frac{303}{1624}$, $\frac{1}{2819449728188377606580651258380126844340006136} a^{15} + \frac{247700090075260480703864153297747247326665}{1409724864094188803290325629190063422170003068} a^{14} + \frac{84910827223515674823760215743853903958011}{402778532598339658082950179768589549191429448} a^{13} + \frac{191344086842681455974959395180890749358087}{2819449728188377606580651258380126844340006136} a^{12} + \frac{80556612405464359742030366769770062152598901}{2819449728188377606580651258380126844340006136} a^{11} + \frac{369965451905670644315979308971287990040274045}{2819449728188377606580651258380126844340006136} a^{10} - \frac{78771766901292354331304982637991313531339175}{352431216023547200822581407297515855542500767} a^{9} + \frac{40946636155750399288769651033035345266405007}{402778532598339658082950179768589549191429448} a^{8} - \frac{680002939812510648735504444674315813711965465}{1409724864094188803290325629190063422170003068} a^{7} - \frac{581198608308068150964925669711941447079369447}{2819449728188377606580651258380126844340006136} a^{6} - \frac{22480421274394536121674689768321267316514108}{50347316574792457260368772471073693648928681} a^{5} + \frac{509371685369116023644404941882195661106070847}{2819449728188377606580651258380126844340006136} a^{4} - \frac{869298091509509227256904775312307563704908015}{2819449728188377606580651258380126844340006136} a^{3} - \frac{696299366658161839502677430307233984415357987}{1409724864094188803290325629190063422170003068} a^{2} + \frac{655785803031931470802720896170743484796314445}{2819449728188377606580651258380126844340006136} a + \frac{1206735282396639803601870293360664543029184025}{2819449728188377606580651258380126844340006136}$

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

Class group and class number

$C_{2}\times C_{6}\times C_{30}$, which has order $360$ (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:  \( 1447320.78419 \) (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

\(\Q(\sqrt{29}) \), \(\Q(\sqrt{377}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{13}, \sqrt{29})\), 4.4.1847677.1 x2, 4.4.63713.1 x2, 4.0.4121741.1, 4.0.24389.1, 8.8.3413910296329.1, 8.0.16988748871081.3, 8.0.2871098559212689.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$13$13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
13.8.6.1$x^{8} - 13 x^{4} + 2704$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$29$29.8.6.1$x^{8} - 203 x^{4} + 68121$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
29.8.6.1$x^{8} - 203 x^{4} + 68121$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$