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Properties

Label 16.0.29702751525...241.24
Degree $16$
Signature $[0, 8]$
Discriminant $23^{8}\cdot 41^{14}$
Root discriminant $123.61$
Ramified primes $23, 41$
Class number $192$ (GRH)
Class group $[2, 4, 24]$ (GRH)
Galois group $C_2^3.C_4$ (as 16T41)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![112126933, -16187548, 7869323, -2448624, 4334218, -3112866, 1287471, 13382, 114839, -92160, 20405, 1550, -680, 130, -13, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 2*x^15 - 13*x^14 + 130*x^13 - 680*x^12 + 1550*x^11 + 20405*x^10 - 92160*x^9 + 114839*x^8 + 13382*x^7 + 1287471*x^6 - 3112866*x^5 + 4334218*x^4 - 2448624*x^3 + 7869323*x^2 - 16187548*x + 112126933)
 
gp: K = bnfinit(x^16 - 2*x^15 - 13*x^14 + 130*x^13 - 680*x^12 + 1550*x^11 + 20405*x^10 - 92160*x^9 + 114839*x^8 + 13382*x^7 + 1287471*x^6 - 3112866*x^5 + 4334218*x^4 - 2448624*x^3 + 7869323*x^2 - 16187548*x + 112126933, 1)
 

Normalized defining polynomial

\( x^{16} - 2 x^{15} - 13 x^{14} + 130 x^{13} - 680 x^{12} + 1550 x^{11} + 20405 x^{10} - 92160 x^{9} + 114839 x^{8} + 13382 x^{7} + 1287471 x^{6} - 3112866 x^{5} + 4334218 x^{4} - 2448624 x^{3} + 7869323 x^{2} - 16187548 x + 112126933 \)

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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{10} a^{9} + \frac{1}{10} a^{8} - \frac{1}{2} a^{6} + \frac{3}{10} a^{5} + \frac{3}{10} a^{4} + \frac{1}{5} a^{3} - \frac{3}{10} a^{2} + \frac{3}{10} a - \frac{1}{5}$, $\frac{1}{10} a^{10} - \frac{1}{10} a^{8} - \frac{1}{2} a^{7} - \frac{1}{5} a^{6} - \frac{1}{10} a^{4} - \frac{1}{2} a^{3} - \frac{2}{5} a^{2} - \frac{1}{2} a + \frac{1}{5}$, $\frac{1}{10} a^{11} + \frac{1}{10} a^{8} + \frac{3}{10} a^{7} - \frac{1}{2} a^{6} - \frac{3}{10} a^{5} + \frac{3}{10} a^{4} + \frac{3}{10} a^{3} + \frac{1}{5} a^{2} + \frac{3}{10}$, $\frac{1}{10} a^{12} + \frac{1}{5} a^{8} - \frac{1}{2} a^{7} + \frac{1}{5} a^{6} + \frac{3}{10} a^{2} + \frac{1}{5}$, $\frac{1}{50} a^{13} + \frac{1}{50} a^{12} - \frac{1}{50} a^{11} - \frac{1}{50} a^{10} + \frac{1}{50} a^{9} - \frac{9}{50} a^{8} - \frac{8}{25} a^{7} - \frac{8}{25} a^{6} - \frac{1}{2} a^{5} + \frac{1}{5} a^{4} + \frac{4}{25} a^{3} - \frac{1}{25} a^{2} - \frac{11}{50} a + \frac{2}{25}$, $\frac{1}{1550} a^{14} + \frac{1}{310} a^{13} + \frac{14}{775} a^{12} + \frac{3}{310} a^{11} + \frac{27}{1550} a^{10} + \frac{293}{1550} a^{8} - \frac{79}{310} a^{7} - \frac{2}{25} a^{6} + \frac{113}{310} a^{5} + \frac{493}{1550} a^{4} - \frac{12}{31} a^{3} + \frac{261}{1550} a^{2} + \frac{21}{62} a + \frac{313}{775}$, $\frac{1}{5629464769422089430512071749573548630320629211876250} a^{15} - \frac{204335191309796140874942484780351260063893127492}{2814732384711044715256035874786774315160314605938125} a^{14} - \frac{172372363112316853595811220909867930513568038957}{22517859077688357722048286998294194521282516847505} a^{13} - \frac{42128399293615158079036685021392941687406519649739}{1125892953884417886102414349914709726064125842375250} a^{12} + \frac{14564918972957692254374968245827979236386630683907}{1125892953884417886102414349914709726064125842375250} a^{11} - \frac{4959159452146157655952150914263281575172248648962}{562946476942208943051207174957354863032062921187625} a^{10} + \frac{700205209359761708660369456878194141909045006917}{30429539294173456381146333781478641244976374118250} a^{9} - \frac{4199478795907450890992396090321756374459975739739}{45035718155376715444096573996588389042565033695010} a^{8} - \frac{2461917660221354044101236047826870795239907389046311}{5629464769422089430512071749573548630320629211876250} a^{7} - \frac{826661443002952441140300779341136903377305846054508}{2814732384711044715256035874786774315160314605938125} a^{6} - \frac{982315739230297871917029800705150913469815820459517}{5629464769422089430512071749573548630320629211876250} a^{5} + \frac{574624802220336650153265235375661131863598749195703}{5629464769422089430512071749573548630320629211876250} a^{4} - \frac{503021907890054629455322217546357045035394290930414}{2814732384711044715256035874786774315160314605938125} a^{3} + \frac{1347010048362521518902109717047768663916279109508061}{2814732384711044715256035874786774315160314605938125} a^{2} - \frac{573332671875591286085752696921987078768931586664553}{2814732384711044715256035874786774315160314605938125} a - \frac{1270785439445522438848481560402286008956515982257131}{5629464769422089430512071749573548630320629211876250}$

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

Class group and class number

$C_{2}\times C_{4}\times C_{24}$, which has order $192$ (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:  \( 117891575.806 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^3.C_4$ (as 16T41):

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

\(\Q(\sqrt{41}) \), 4.0.36459209.2, 4.2.38663.1, 4.2.1585183.1, 8.0.54500230757132921.4 x2, 8.0.1329273920905681.4

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}$ R ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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}$