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Properties

Label 16.0.92383372603...1376.5
Degree $16$
Signature $[0, 8]$
Discriminant $2^{48}\cdot 3^{8}\cdot 29^{8}$
Root discriminant $74.62$
Ramified primes $2, 3, 29$
Class number $47736$ (GRH)
Class group $[3, 3, 5304]$ (GRH)
Galois group $C_4\times C_2^2$ (as 16T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![74638654, -6189912, 41438996, -847320, 10592826, 74400, 1694252, 22368, 185223, 1512, 14660, -24, 854, 0, 36, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 + 36*x^14 + 854*x^12 - 24*x^11 + 14660*x^10 + 1512*x^9 + 185223*x^8 + 22368*x^7 + 1694252*x^6 + 74400*x^5 + 10592826*x^4 - 847320*x^3 + 41438996*x^2 - 6189912*x + 74638654)
 
gp: K = bnfinit(x^16 + 36*x^14 + 854*x^12 - 24*x^11 + 14660*x^10 + 1512*x^9 + 185223*x^8 + 22368*x^7 + 1694252*x^6 + 74400*x^5 + 10592826*x^4 - 847320*x^3 + 41438996*x^2 - 6189912*x + 74638654, 1)
 

Normalized defining polynomial

\( x^{16} + 36 x^{14} + 854 x^{12} - 24 x^{11} + 14660 x^{10} + 1512 x^{9} + 185223 x^{8} + 22368 x^{7} + 1694252 x^{6} + 74400 x^{5} + 10592826 x^{4} - 847320 x^{3} + 41438996 x^{2} - 6189912 x + 74638654 \)

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:  \(923833726039318399509778661376=2^{48}\cdot 3^{8}\cdot 29^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $74.62$
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:  $2, 3, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1392=2^{4}\cdot 3\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{1392}(1,·)$, $\chi_{1392}(1217,·)$, $\chi_{1392}(521,·)$, $\chi_{1392}(407,·)$, $\chi_{1392}(1103,·)$, $\chi_{1392}(1045,·)$, $\chi_{1392}(755,·)$, $\chi_{1392}(463,·)$, $\chi_{1392}(349,·)$, $\chi_{1392}(869,·)$, $\chi_{1392}(811,·)$, $\chi_{1392}(173,·)$, $\chi_{1392}(115,·)$, $\chi_{1392}(697,·)$, $\chi_{1392}(59,·)$, $\chi_{1392}(1159,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{384440777} a^{14} - \frac{35932045}{384440777} a^{13} - \frac{157164951}{384440777} a^{12} + \frac{18681074}{384440777} a^{11} - \frac{72154252}{384440777} a^{10} - \frac{163010311}{384440777} a^{9} - \frac{1517673}{54920111} a^{8} + \frac{5842180}{384440777} a^{7} - \frac{169179030}{384440777} a^{6} + \frac{4917854}{384440777} a^{5} + \frac{85209917}{384440777} a^{4} + \frac{2710114}{8179591} a^{3} - \frac{50329648}{384440777} a^{2} - \frac{3184541}{8179591} a + \frac{73152364}{384440777}$, $\frac{1}{304500876217272726269924910211276889} a^{15} - \frac{30296684103446779763731893}{43500125173896103752846415744468127} a^{14} + \frac{90909807180869837601110082921374591}{304500876217272726269924910211276889} a^{13} - \frac{7850620330357650094532539405398640}{304500876217272726269924910211276889} a^{12} - \frac{23245587411471210979173540696630043}{304500876217272726269924910211276889} a^{11} - \frac{122638610996399362084067667006961920}{304500876217272726269924910211276889} a^{10} - \frac{92474428427029347905449581062702261}{304500876217272726269924910211276889} a^{9} - \frac{120351049185048343132605186216719736}{304500876217272726269924910211276889} a^{8} + \frac{120944632092848064843123392778624012}{304500876217272726269924910211276889} a^{7} + \frac{6042417534020174092122661781884828}{17911816248074866251172053541839817} a^{6} - \frac{65084497458573236885462981976584400}{304500876217272726269924910211276889} a^{5} + \frac{106386854498764388620765321890583939}{304500876217272726269924910211276889} a^{4} - \frac{1895897207567112036753394712870845}{17911816248074866251172053541839817} a^{3} + \frac{126993886462559827320068333217786236}{304500876217272726269924910211276889} a^{2} - \frac{3993183055761293290869905108539071}{13239168531185770707388039574403343} a - \frac{76797412859126772838511462960157570}{304500876217272726269924910211276889}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{5304}$, which has order $47736$ (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:  \( 11964.310642723332 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_4$ (as 16T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 16
The 16 conjugacy class representatives for $C_4\times C_2^2$
Character table for $C_4\times C_2^2$

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{-174}) \), \(\Q(\sqrt{-58}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-87}) \), \(\Q(\sqrt{-29}) \), \(\Q(\sqrt{3}, \sqrt{-58})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\sqrt{3}, \sqrt{-29})\), \(\Q(\sqrt{2}, \sqrt{-87})\), \(\Q(\sqrt{6}, \sqrt{-29})\), \(\Q(\sqrt{2}, \sqrt{-29})\), \(\Q(\sqrt{6}, \sqrt{-58})\), 4.0.15501312.6, 4.0.1722368.6, \(\Q(\zeta_{16})^+\), 4.4.18432.1, 8.0.3754541776896.5, 8.0.961162694885376.31, \(\Q(\zeta_{48})^+\), 8.0.240290673721344.80, 8.0.240290673721344.39, 8.0.961162694885376.28, 8.0.11866206109696.2

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 R R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/31.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{16}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}$

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
$2$2.8.24.9$x^{8} + 8 x^{7} + 14 x^{4} + 4 x^{2} + 8 x + 30$$8$$1$$24$$C_4\times C_2$$[2, 3, 4]$
2.8.24.9$x^{8} + 8 x^{7} + 14 x^{4} + 4 x^{2} + 8 x + 30$$8$$1$$24$$C_4\times C_2$$[2, 3, 4]$
$3$3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3.8.4.1$x^{8} + 36 x^{4} - 27 x^{2} + 324$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$29$29.8.4.1$x^{8} + 31958 x^{4} - 24389 x^{2} + 255328441$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
29.8.4.1$x^{8} + 31958 x^{4} - 24389 x^{2} + 255328441$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$