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Properties

Label 16.0.86380562306...0625.1
Degree $16$
Signature $[0, 8]$
Discriminant $5^{12}\cdot 29^{12}$
Root discriminant $41.79$
Ramified primes $5, 29$
Class number $8$ (GRH)
Class group $[2, 4]$ (GRH)
Galois group $C_4^2$ (as 16T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![279841, -243340, 162932, -111527, 67057, -36968, 18249, -7506, 2045, 341, 482, -241, 22, -13, -3, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - x^15 - 3*x^14 - 13*x^13 + 22*x^12 - 241*x^11 + 482*x^10 + 341*x^9 + 2045*x^8 - 7506*x^7 + 18249*x^6 - 36968*x^5 + 67057*x^4 - 111527*x^3 + 162932*x^2 - 243340*x + 279841)
 
gp: K = bnfinit(x^16 - x^15 - 3*x^14 - 13*x^13 + 22*x^12 - 241*x^11 + 482*x^10 + 341*x^9 + 2045*x^8 - 7506*x^7 + 18249*x^6 - 36968*x^5 + 67057*x^4 - 111527*x^3 + 162932*x^2 - 243340*x + 279841, 1)
 

Normalized defining polynomial

\( x^{16} - x^{15} - 3 x^{14} - 13 x^{13} + 22 x^{12} - 241 x^{11} + 482 x^{10} + 341 x^{9} + 2045 x^{8} - 7506 x^{7} + 18249 x^{6} - 36968 x^{5} + 67057 x^{4} - 111527 x^{3} + 162932 x^{2} - 243340 x + 279841 \)

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:  \(86380562306022715087890625=5^{12}\cdot 29^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.79$
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:  $5, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(145=5\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{145}(128,·)$, $\chi_{145}(1,·)$, $\chi_{145}(133,·)$, $\chi_{145}(12,·)$, $\chi_{145}(144,·)$, $\chi_{145}(17,·)$, $\chi_{145}(86,·)$, $\chi_{145}(88,·)$, $\chi_{145}(28,·)$, $\chi_{145}(99,·)$, $\chi_{145}(104,·)$, $\chi_{145}(41,·)$, $\chi_{145}(46,·)$, $\chi_{145}(117,·)$, $\chi_{145}(57,·)$, $\chi_{145}(59,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a$, $\frac{1}{14} a^{12} - \frac{1}{7} a^{11} - \frac{1}{7} a^{10} + \frac{2}{7} a^{9} - \frac{3}{14} a^{7} + \frac{2}{7} a^{5} + \frac{2}{7} a^{3} - \frac{1}{14} a^{2} + \frac{3}{7} a + \frac{1}{7}$, $\frac{1}{396363701062} a^{13} + \frac{438279409}{198181850531} a^{12} + \frac{35731383979}{396363701062} a^{11} - \frac{39237619271}{396363701062} a^{10} - \frac{60609546514}{198181850531} a^{9} - \frac{8286604875}{396363701062} a^{8} - \frac{11681947262}{198181850531} a^{7} - \frac{16753725117}{396363701062} a^{6} - \frac{106929068139}{396363701062} a^{5} + \frac{19543184544}{198181850531} a^{4} + \frac{46712992075}{396363701062} a^{3} + \frac{12605700122}{198181850531} a^{2} - \frac{145754097743}{396363701062} a + \frac{8412203111}{17233204394}$, $\frac{1}{9116365124426} a^{14} - \frac{1}{9116365124426} a^{13} - \frac{101919490879}{9116365124426} a^{12} - \frac{53998768067}{1302337874918} a^{11} + \frac{392834669991}{4558182562213} a^{10} + \frac{1825745387187}{9116365124426} a^{9} + \frac{377370510269}{9116365124426} a^{8} - \frac{2335227330085}{9116365124426} a^{7} + \frac{1669256904655}{9116365124426} a^{6} + \frac{1856961963567}{4558182562213} a^{5} + \frac{1250617314537}{9116365124426} a^{4} - \frac{339405086431}{9116365124426} a^{3} + \frac{452594238883}{1302337874918} a^{2} - \frac{81506971437}{396363701062} a + \frac{3489022350}{8616602197}$, $\frac{1}{209676397861798} a^{15} - \frac{1}{209676397861798} a^{14} - \frac{3}{209676397861798} a^{13} - \frac{597654008557}{104838198930899} a^{12} + \frac{28289540379}{104838198930899} a^{11} + \frac{8730130039055}{104838198930899} a^{10} + \frac{888511991601}{29953771123114} a^{9} - \frac{49700121475185}{209676397861798} a^{8} - \frac{4848507013559}{14976885561557} a^{7} + \frac{10689118027606}{104838198930899} a^{6} + \frac{29115994090655}{104838198930899} a^{5} + \frac{50499448843839}{209676397861798} a^{4} + \frac{11521544245487}{209676397861798} a^{3} + \frac{1402400403296}{4558182562213} a^{2} + \frac{29346685980}{198181850531} a - \frac{163382991}{17233204394}$

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

Class group and class number

$C_{2}\times C_{4}$, which has order $8$ (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:  \( \frac{13147}{28311692933} a^{15} + \frac{39441}{28311692933} a^{14} + \frac{170911}{28311692933} a^{13} - \frac{289234}{28311692933} a^{12} - \frac{3138961}{56623385866} a^{11} - \frac{6336854}{28311692933} a^{10} - \frac{4483127}{28311692933} a^{9} - \frac{26885615}{28311692933} a^{8} + \frac{98681382}{28311692933} a^{7} + \frac{295959265}{56623385866} a^{6} + \frac{486018296}{28311692933} a^{5} - \frac{881598379}{28311692933} a^{4} + \frac{63749803}{1230943171} a^{3} - \frac{93133348}{1230943171} a^{2} + \frac{37214858433}{56623385866} a - \frac{159959549}{1230943171} \) (order $10$)
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:  \( 805904.787678 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4^2$ (as 16T4):

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 $C_4^2$
Character table for $C_4^2$

Intermediate fields

\(\Q(\sqrt{145}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{29}) \), 4.4.3048625.2, \(\Q(\sqrt{5}, \sqrt{29})\), 4.4.3048625.1, 4.0.105125.2, \(\Q(\zeta_{5})\), 4.0.609725.2, 4.0.24389.1, 8.8.9294114390625.2, 8.0.11051265625.1, 8.0.371764575625.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}$ R ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$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}$