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Properties

Label 16.8.34182052039...7929.2
Degree $16$
Signature $[8, 4]$
Discriminant $37^{14}\cdot 41^{14}$
Root discriminant $607.24$
Ramified primes $37, 41$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $C_8.C_4$ (as 16T49)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3644035361, 23032733237, -10796424454, 1250599119, -384839693, 113723601, -5402229, -29022674, 10492820, -772381, -115080, 59103, -7321, 63, -13, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 4*x^15 - 13*x^14 + 63*x^13 - 7321*x^12 + 59103*x^11 - 115080*x^10 - 772381*x^9 + 10492820*x^8 - 29022674*x^7 - 5402229*x^6 + 113723601*x^5 - 384839693*x^4 + 1250599119*x^3 - 10796424454*x^2 + 23032733237*x + 3644035361)
 
gp: K = bnfinit(x^16 - 4*x^15 - 13*x^14 + 63*x^13 - 7321*x^12 + 59103*x^11 - 115080*x^10 - 772381*x^9 + 10492820*x^8 - 29022674*x^7 - 5402229*x^6 + 113723601*x^5 - 384839693*x^4 + 1250599119*x^3 - 10796424454*x^2 + 23032733237*x + 3644035361, 1)
 

Normalized defining polynomial

\( x^{16} - 4 x^{15} - 13 x^{14} + 63 x^{13} - 7321 x^{12} + 59103 x^{11} - 115080 x^{10} - 772381 x^{9} + 10492820 x^{8} - 29022674 x^{7} - 5402229 x^{6} + 113723601 x^{5} - 384839693 x^{4} + 1250599119 x^{3} - 10796424454 x^{2} + 23032733237 x + 3644035361 \)

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:  \(341820520390299824133982419608164514148827929=37^{14}\cdot 41^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $607.24$
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:  $37, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{41} a^{8} - \frac{2}{41} a^{7} + \frac{12}{41} a^{6} - \frac{6}{41} a^{5} + \frac{7}{41} a^{4} + \frac{15}{41} a^{3} - \frac{7}{41} a^{2} - \frac{20}{41} a + \frac{16}{41}$, $\frac{1}{82} a^{9} - \frac{1}{82} a^{8} - \frac{31}{82} a^{7} + \frac{3}{41} a^{6} - \frac{20}{41} a^{5} + \frac{11}{41} a^{4} - \frac{33}{82} a^{3} - \frac{27}{82} a^{2} - \frac{2}{41} a - \frac{25}{82}$, $\frac{1}{164} a^{10} - \frac{1}{82} a^{8} + \frac{79}{164} a^{7} - \frac{1}{82} a^{6} + \frac{12}{41} a^{5} - \frac{47}{164} a^{4} + \frac{31}{82} a^{3} - \frac{77}{164} a^{2} + \frac{27}{164} a - \frac{37}{164}$, $\frac{1}{27224} a^{11} + \frac{71}{27224} a^{10} - \frac{81}{13612} a^{9} + \frac{201}{27224} a^{8} + \frac{7899}{27224} a^{7} - \frac{5725}{13612} a^{6} - \frac{11691}{27224} a^{5} + \frac{10333}{27224} a^{4} + \frac{6245}{27224} a^{3} + \frac{466}{3403} a^{2} + \frac{1121}{3403} a - \frac{9919}{27224}$, $\frac{1}{108896} a^{12} + \frac{109}{108896} a^{10} + \frac{5}{1312} a^{9} - \frac{265}{27224} a^{8} + \frac{24657}{108896} a^{7} - \frac{10813}{108896} a^{6} + \frac{10841}{54448} a^{5} + \frac{16109}{54448} a^{4} - \frac{22011}{108896} a^{3} + \frac{991}{6806} a^{2} - \frac{6551}{108896} a + \frac{20329}{108896}$, $\frac{1}{217792} a^{13} - \frac{1}{217792} a^{12} - \frac{3}{217792} a^{11} + \frac{161}{108896} a^{10} + \frac{733}{217792} a^{9} - \frac{2107}{217792} a^{8} + \frac{52529}{108896} a^{7} - \frac{58257}{217792} a^{6} + \frac{10611}{27224} a^{5} - \frac{13669}{217792} a^{4} + \frac{60859}{217792} a^{3} + \frac{67353}{217792} a^{2} - \frac{749}{3403} a + \frac{20231}{217792}$, $\frac{1}{188607872} a^{14} - \frac{135}{94303936} a^{13} + \frac{97}{94303936} a^{12} - \frac{119}{188607872} a^{11} - \frac{22981}{188607872} a^{10} - \frac{236491}{47151968} a^{9} + \frac{2053729}{188607872} a^{8} - \frac{18427891}{188607872} a^{7} + \frac{60858205}{188607872} a^{6} + \frac{45839859}{188607872} a^{5} - \frac{5182741}{47151968} a^{4} - \frac{8047895}{94303936} a^{3} - \frac{88458373}{188607872} a^{2} - \frac{28333441}{188607872} a + \frac{78794621}{188607872}$, $\frac{1}{79652801471184142837959762890273505763965901568} a^{15} + \frac{191786112433431192119637943623245107535}{79652801471184142837959762890273505763965901568} a^{14} + \frac{4167178741167260414513406154044296273909}{19913200367796035709489940722568376440991475392} a^{13} - \frac{318315188606069046042888060012130039517133}{79652801471184142837959762890273505763965901568} a^{12} + \frac{164908785510697229342977320157441721603841}{9956600183898017854744970361284188220495737696} a^{11} + \frac{778532104893489481393252423651093307170515}{1942751255394735191169750314396914774730875648} a^{10} + \frac{98552812470520499371222590089450418776013717}{79652801471184142837959762890273505763965901568} a^{9} + \frac{5284377761746474185112536468776866811464793}{39826400735592071418979881445136752881982950784} a^{8} + \frac{3129195077404529775498225885680312210932052031}{39826400735592071418979881445136752881982950784} a^{7} - \frac{5616704818946973084636931071138172210303631673}{19913200367796035709489940722568376440991475392} a^{6} - \frac{22183982574144395566064677887144549861947479709}{79652801471184142837959762890273505763965901568} a^{5} + \frac{7065984223740862212810541798273676033350867291}{39826400735592071418979881445136752881982950784} a^{4} - \frac{5986502718474419639531833913969669954148027723}{79652801471184142837959762890273505763965901568} a^{3} + \frac{8195697636586175166626269301229778817944630891}{39826400735592071418979881445136752881982950784} a^{2} - \frac{280049103822211190032111916511913187972132593}{9956600183898017854744970361284188220495737696} a - \frac{6117968340758329539777029976745877745176321447}{79652801471184142837959762890273505763965901568}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  \( 3132062581460000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_8.C_4$ (as 16T49):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_8.C_4$
Character table for $C_8.C_4$

Intermediate fields

\(\Q(\sqrt{1517}) \), \(\Q(\sqrt{41}) \), \(\Q(\sqrt{37}) \), \(\Q(\sqrt{37}, \sqrt{41})\), 8.8.12187467896636600569.1

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

Sibling fields

Galois closure: 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}$ R R ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}$

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
$37$37.8.7.1$x^{8} - 37$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
37.8.7.1$x^{8} - 37$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
$41$41.8.7.2$x^{8} - 1476$$8$$1$$7$$C_8$$[\ ]_{8}$
41.8.7.4$x^{8} - 1912896$$8$$1$$7$$C_8$$[\ ]_{8}$