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Properties

Label 16.0.29702751525...241.29
Degree $16$
Signature $[0, 8]$
Discriminant $23^{8}\cdot 41^{14}$
Root discriminant $123.61$
Ramified primes $23, 41$
Class number $384$ (GRH)
Class group $[2, 4, 48]$ (GRH)
Galois group $C_2^3.C_4$ (as 16T36)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![534849200, 125046840, 149722596, 112760966, 31633951, 13465847, 7635373, 556314, 619392, 40593, 46905, -9518, 1922, -444, 68, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 6*x^15 + 68*x^14 - 444*x^13 + 1922*x^12 - 9518*x^11 + 46905*x^10 + 40593*x^9 + 619392*x^8 + 556314*x^7 + 7635373*x^6 + 13465847*x^5 + 31633951*x^4 + 112760966*x^3 + 149722596*x^2 + 125046840*x + 534849200)
 
gp: K = bnfinit(x^16 - 6*x^15 + 68*x^14 - 444*x^13 + 1922*x^12 - 9518*x^11 + 46905*x^10 + 40593*x^9 + 619392*x^8 + 556314*x^7 + 7635373*x^6 + 13465847*x^5 + 31633951*x^4 + 112760966*x^3 + 149722596*x^2 + 125046840*x + 534849200, 1)
 

Normalized defining polynomial

\( x^{16} - 6 x^{15} + 68 x^{14} - 444 x^{13} + 1922 x^{12} - 9518 x^{11} + 46905 x^{10} + 40593 x^{9} + 619392 x^{8} + 556314 x^{7} + 7635373 x^{6} + 13465847 x^{5} + 31633951 x^{4} + 112760966 x^{3} + 149722596 x^{2} + 125046840 x + 534849200 \)

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:  \(2970275152580737243393920061992241=23^{8}\cdot 41^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $123.61$
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:  $23, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{460} a^{12} + \frac{5}{23} a^{11} + \frac{111}{460} a^{10} + \frac{27}{230} a^{9} + \frac{179}{460} a^{8} - \frac{1}{2} a^{7} + \frac{1}{46} a^{6} - \frac{189}{460} a^{5} - \frac{53}{115} a^{4} + \frac{11}{92} a^{3} + \frac{99}{460} a^{2} + \frac{71}{230} a - \frac{1}{23}$, $\frac{1}{2300} a^{13} + \frac{1}{2300} a^{11} - \frac{233}{1150} a^{10} - \frac{37}{100} a^{9} + \frac{27}{230} a^{8} - \frac{11}{115} a^{7} - \frac{729}{2300} a^{6} - \frac{273}{575} a^{5} - \frac{27}{460} a^{4} + \frac{1039}{2300} a^{3} + \frac{148}{575} a^{2} - \frac{111}{230} a - \frac{3}{23}$, $\frac{1}{1761137485000} a^{14} + \frac{48224519}{1761137485000} a^{13} + \frac{1085457781}{1761137485000} a^{12} - \frac{130490440197}{1761137485000} a^{11} + \frac{3352638683}{14089099880} a^{10} - \frac{758829522829}{1761137485000} a^{9} + \frac{74722575873}{176113748500} a^{8} - \frac{262610039159}{1761137485000} a^{7} + \frac{530791737657}{1761137485000} a^{6} + \frac{123988199397}{1761137485000} a^{5} - \frac{19089687392}{220142185625} a^{4} - \frac{518349878467}{1761137485000} a^{3} - \frac{88200901273}{440284371250} a^{2} + \frac{15161635491}{88056874250} a + \frac{228299208}{8805687425}$, $\frac{1}{2048258501076903309051407991927944694781995845000} a^{15} + \frac{54857974913787688872078296341763976}{256032312634612913631425998990993086847749480625} a^{14} - \frac{57105924495983245755179317182524625799823507}{512064625269225827262851997981986173695498961250} a^{13} + \frac{526137480571452670230244086953573823316379131}{1024129250538451654525703995963972347390997922500} a^{12} + \frac{127434865431868178905469621425720129977007686821}{1024129250538451654525703995963972347390997922500} a^{11} - \frac{113045892302038464956322953605093378154420327851}{512064625269225827262851997981986173695498961250} a^{10} - \frac{11337989024064947361116351855258667692548167937}{89054717438126230828322086605562812816608515000} a^{9} - \frac{960933700704425818727601703889952199801700759939}{2048258501076903309051407991927944694781995845000} a^{8} - \frac{6608081727857382044229363318450860249257284781}{27679168933471666338532540431458712091648592500} a^{7} - \frac{67624306540975811154011564677803502302812639303}{204825850107690330905140799192794469478199584500} a^{6} + \frac{1014324393671308098262894507151900595699422974447}{2048258501076903309051407991927944694781995845000} a^{5} - \frac{8344634925914078120702294367178957311015972183}{55358337866943332677065080862917424183297185000} a^{4} - \frac{9251556311448859763800980281004428310461082201}{409651700215380661810281598385588938956399169000} a^{3} - \frac{113230675421362699237867815941349280234121956059}{1024129250538451654525703995963972347390997922500} a^{2} - \frac{3015061609554054705839685867067997506762288423}{51206462526922582726285199798198617369549896125} a + \frac{33589104572023392280244238214229905381364752}{330364274367242469201839998698055595932579975}$

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

Class group and class number

$C_{2}\times C_{4}\times C_{48}$, which has order $384$ (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:  \( 31804349.3331 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_4$ (as 16T36):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 11 conjugacy class representatives for $C_2^3.C_4$
Character table for $C_2^3.C_4$

Intermediate fields

\(\Q(\sqrt{41}) \), \(\Q(\sqrt{-23}) \), \(\Q(\sqrt{-943}) \), 4.0.36459209.2, 4.4.68921.1, \(\Q(\sqrt{-23}, \sqrt{41})\), 8.0.54500230757132921.4 x2, 8.4.103025010883049.2 x2, 8.0.1329273920905681.3

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 8 siblings: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$23$23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$41$41.8.7.3$x^{8} - 53136$$8$$1$$7$$C_8$$[\ ]_{8}$
41.8.7.3$x^{8} - 53136$$8$$1$$7$$C_8$$[\ ]_{8}$